JEE MAIN · MATHEMATICS · CONIC SECTIONS
JEE Main Conic Sections Questions & Solutions
146 solved questions on Conic Sections, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
146 solved questions on Conic Sections, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
If the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ meets the line $\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1$ on the $x$-axis and the line $\frac{x}{7}-\frac{y}{2 \sqrt{6}}=1$ on…
View solution →If the common tangent to the parabolas, y2 = 4x and x2 = 4y also touches the circle, x2 + y2 = c2, then c is equal to :
View solution →If two tangents drawn from a point P to the parabola y2 = 16(x $-$ 3) are at right angles, then the locus of point P is :
View solution →Let the foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$ and the hyperbola $\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}$ coincide. Then the length of the latus…
View solution →If $y = {m_1}x + {c_1}$ and $y = {m_2}x + {c_2}$, ${m_1} \ne {m_2}$ are two common tangents of circle ${x^2} + {y^2} = 2$ and parabola y2 = x, then the value of $8|{m_1}{m_2}|$ is…
View solution →The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid point is $\left(1, \frac{2}{5}\right)$, is equal to :
View solution →If the line y = mx + c is a common tangent to the hyperbola ${{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1$ and the circle x2 + y2 = 36, then which one of the following is true?
View solution →For the hyperbola $\mathrm{H}: x^{2}-y^{2}=1$ and the ellipse $\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$, a $\mathrm{b}0$, let the (1) eccentricity…
View solution →Let the tangent to the curve $x^{2}+2 x-4 y+9=0$ at the point $\mathrm{P}(1,3)$ on it meet the $y$-axis at $\mathrm{A}$. Let the line passing through $\mathrm{P}$ and parallel to…
View solution →Let $$\mathrm{P}\left(\frac{2 \sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right), \mathrm{Q}, \mathrm{R}$$ and $\mathrm{S}$ be four points on the ellipse $9 x^{2}+4 y^{2}=36$. Let…
View solution →If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to :
View solution →Let a line perpendicular to the line $2 x-y=10$ touch the parabola $y^2=4(x-9)$ at the point P. The distance of the point P from the centre of the circle $x^2+y^2-14 x-8 y+56=0$…
View solution →If e1 and e2 are the eccentricities of the ellipse, ${{{x^2}} \over {18}} + {{{y^2}} \over 4} = 1$ and the hyperbola, ${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$ respectively and…
View solution →Let $m_{1}$ and $m_{2}$ be the slopes of the tangents drawn from the point $\mathrm{P}(4,1)$ to the hyperbola $H: \frac{y^{2}}{25}-\frac{x^{2}}{16}=1$. If $\mathrm{Q}$ is the…
View solution →Let a line L1 be tangent to the hyperbola ${{{x^2}} \over {16}} - {{{y^2}} \over 4} = 1$ and let L2 be the line passing through the origin and perpendicular to L1. If the locus of…
View solution →If the shortest distance of the parabola $y^2=4 x$ from the centre of the circle $x^2+y^2-4 x-16 y+64=0$ is $\mathrm{d}$, then $\mathrm{d}^2$ is equal to :
View solution →If the length of the latus rectum of a parabola, whose focus is $(a, a)$ and the tangent at its vertex is $x+y=a$, is 16, then $|a|$ is equal to :
View solution →The foci of a hyperbola are $( \pm 2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line $2 x+3 y=6$, is drawn at a point in the first quadrant on the…
View solution →The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse ${x^2} + 2{y^2} = 4$ is an ellipse with eccentricity :
View solution →If two tangents drawn from a point ($\alpha$, $\beta$) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other,…
View solution →Let $P Q$ be a chord of the parabola $y^2=12 x$ and the midpoint of $P Q$ be at $(4,1)$. Then, which of the following point lies on the line passing through the points…
View solution →If the line $y = 4 + kx,\,k 0$, is the tangent to the parabola $y = x - {x^2}$ at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :
View solution →The line $12x\cos \theta + 5y\sin \theta = 60$ is tangent to which of the following curves?
View solution →Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16 x$. If one of these tangents touches the two curves at $Q$ and $R$,…
View solution →If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies :
View solution →If the co-ordinates of two points A and B are $\left( {\sqrt 7 ,0} \right)$ and $\left( { - \sqrt 7 ,0} \right)$ respectively and P is any point on the conic, 9x2 + 16y2 = 144,…
View solution →If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $3x + y - 29 = 0$, is ${x^2} + a{y^2} + bxy + cx + dy + k = 0$, then $a + b + c + d + k$ is equal…
View solution →The length of the minor axis (along y-axis) of an ellipse in the standard form is ${4 \over {\sqrt 3 }}$. If this ellipse touches the line, x + 6y = 8; then its eccentricity is :
View solution →Let the hyperbola $H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ pass through the point $(2 \sqrt{2},-2 \sqrt{2})$. A parabola is drawn whose focus is same as the focus of…
View solution →Let $\mathrm{P}\left(x_{0}, y_{0}\right)$ be the point on the hyperbola $3 x^{2}-4 y^{2}=36$, which is nearest to the line $3 x+2 y=1$. Then $\sqrt{2}\left(y_{0}-x_{0}\right)$ is…
View solution →The locus of the point of intersection of the lines $\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0$ and $\sqrt 3 x - y - 4\left( {\sqrt 3 } \right)k = 0$ is a conic, whose…
View solution →Let the ellipse $3x^2 + py^2 = 4$ pass through the centre $C$ of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ of radius $r$. Let $f_1, f_2$ be the focal distances of the point $C$ on…
View solution →The normal to the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over 9} = 1$ at the point $\left( {8,3\sqrt 3 } \right)$ on it passes through the point :
View solution →Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a}\mathrm{b}$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latusrectum is $\sqrt{14}$. Then the…
View solution →A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the…
View solution →Let a tangent to the curve $9{x^2} + 16{y^2} = 144$ intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is ________
View solution →Let $\mathrm{y}=f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y=-\frac{1}{2}$. Then $$S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin…
View solution →If the points of intersections of the ellipse ${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$ and the circle x2 + y2 = 4b, b > 4 lie on the curve y2 = 3x2, then b is equal to…
View solution →Let the line $\mathrm{L}: \sqrt{2} x+y=\alpha$ pass through the point of the intersection $\mathrm{P}$ (in the first quadrant) of the circle $x^2+y^2=3$ and the parabola $x^2=2…
View solution →Let a tangent be drawn to the ellipse ${{{x^2}} \over {27}} + {y^2} = 1$ at $(3\sqrt 3 \cos \theta ,\sin \theta )$ where $0 \in \left( {0,{\pi \over 2}} \right)$. Then the value…
View solution →If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is :
View solution →Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 $-$ y2 = 3. If L is also a tangent to the parabola y2 = $\alpha$x, then $\alpha$ is equal to :
View solution →For some $\theta \in \left( {0,{\pi \over 2}} \right)$, if the eccentricity of the hyperbola, x2–y2sec2$\theta$ = 10 is $\sqrt 5$ times the eccentricity of the ellipse,…
View solution →Let $P(a, b)$ be a point on the parabola $y^{2}=8 x$ such that the tangent at $P$ passes through the centre of the circle $x^{2}+y^{2}-10 x-14 y+65=0$. Let $A$ be the product of…
View solution →Let y = mx + c, m > 0 be the focal chord of y2 = $-$ 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of 4$\sqrt 2$ (m + c) is equal to _____________.
View solution →The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, ${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1$ is :
View solution →If 3x + 4y = 12$\sqrt 2$ is a tangent to the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1$ for some $a$ $\in$ R, then the distance between the foci of the ellipse is :
View solution →Let $f(x)=x^2+9, g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f \circ g(10), \mathrm{b}=g \circ f(3)$. If $\mathrm{e}$ and $l$ denote the eccentricity and the length of the latus rectum of…
View solution →If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the…
View solution →Let $y^2=12 x$ be the parabola and $S$ be its focus. Let $P Q$ be a focal chord of the parabola such that $(S P)(S Q)=\frac{147}{4}$. Let $C$ be the circle described taking $P Q$…
View solution →The locus of mid-points of the line segments joining ($-$3, $-$5) and the points on the ellipse ${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$ is :
View solution →Let a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half…
View solution →If vertex of a parabola is (2, $-$1) and the equation of its directrix is 4x $-$ 3y = 21, then the length of its latus rectum is :
View solution →The parabolas : $a x^2+2 b x+c y=0$ and $d x^2+2 e x+f y=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then :
View solution →The locus of the mid points of the chords of the hyperbola x2 $-$ y2 = 4, which touch the parabola y2 = 8x, is :
View solution →Let $\mathrm{H}$ be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is :
View solution →Let $\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}\mathrm{b}$ and $\mathrm{H}: \frac{x^2}{\mathrm{~A}^2}-\frac{y^2}{\mathrm{~B}^2}=1$. Let the…
View solution →If the line $x-1=0$ is a directrix of the hyperbola $k x^{2}-y^{2}=6$, then the hyperbola passes through the point :
View solution →Let the tangent drawn to the parabola $y^{2}=24 x$ at the point $(\alpha, \beta)$ is perpendicular to the line $2 x+2 y=5$. Then the normal to the hyperbola…
View solution →Let one focus of the hyperbola $\mathrm{H}: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ be at $(\sqrt{10}, 0)$ and the corresponding directrix be…
View solution →Let L be a common tangent line to the curves 4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31. Then the square of the slope of the line L is __________.
View solution →The line y = x + 1 meets the ellipse ${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$ at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)2 is equal to…
View solution →Suppose $\mathrm{AB}$ is a focal chord of the parabola $y^2=12 x$ of length $l$ and slope $\mathrm{m}
View solution →The centre of the circle passing through the point (0, 1) and touching the parabola y = x2 at the point (2, 4) is :
View solution →Let $H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$, a 0, b 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2\sqrt 2 +…
View solution →Let the common tangents to the curves $4({x^2} + {y^2}) = 9$ and ${y^2} = 4x$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and…
View solution →Let the hyperbola $H:{{{x^2}} \over {{a^2}}} - {y^2} = 1$ and the ellipse $E:3{x^2} + 4{y^2} = 12$ be such that the length of latus rectum of H is equal to the length of latus…
View solution →An ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the vertices of the hyperbola $H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$. Let the major and minor axes of…
View solution →If the $x$-intercept of a focal chord of the parabola $y^{2}=8x+4y+4$ is 3, then the length of this chord is equal to ____________.
View solution →A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it?
View solution →Let $\mathrm{H}_{\mathrm{n}}: \frac{x^{2}}{1+n}-\frac{y^{2}}{3+n}=1, n \in N$. Let $\mathrm{k}$ be the smallest even value of $\mathrm{n}$ such that the eccentricity of…
View solution →If $\mathrm{P}(\mathrm{h}, \mathrm{k})$ be a point on the parabola $x=4 y^{2}$, which is nearest to the point $\mathrm{Q}(0,33)$, then the distance of $\mathrm{P}$ from the…
View solution →Let the line y = mx and the ellipse 2x2 + y2 = 1 intersect at a ponit P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at $\left( { - {1…
View solution →Let $\theta$ be the acute angle between the tangents to the ellipse ${{{x^2}} \over 9} + {{{y^2}} \over 1} = 1$ and the circle ${x^2} + {y^2} = 3$ at their point of intersection…
View solution →Let ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the…
View solution →Let P(3, 3) be a point on the hyperbola, ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$. If the normal to it at P intersects the x-axis at (9, 0) and e is its…
View solution →A hyperbola having the transverse axis of length $\sqrt 2$ has the same foci as that of the ellipse 3x2 + 4y2 = 12, then this hyperbola does not pass through which of the…
View solution →Let a circle of radius 4 be concentric to the ellipse $15 x^{2}+19 y^{2}=285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle :
View solution →Let L be a tangent line to the parabola y2 = 4x $-$ 20 at (6, 2). If L is also a tangent to the ellipse ${{{x^2}} \over 2} + {{{y^2}} \over b} = 1$, then the value of b is equal…
View solution →A tangent and a normal are drawn at the point P(2, $-$4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a…
View solution →The line $x=8$ is the directrix of the ellipse $\mathrm{E}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with the corresponding focus $(2,0)$. If the tangent to $\mathrm{E}$ at the…
View solution →The distance of the point $(6,-2\sqrt2)$ from the common tangent $\mathrm{y=mx+c,m 0}$, of the curves $x=2y^2$ and $x=1+y^2$ is :
View solution →Let e1 and e2 be the eccentricities of the ellipse, ${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$(b < 5) and the hyperbola, ${{{x^2}} \over {16}} - {{{y^2}} \over {{b^2}}}…
View solution →If the tangent at a point P on the parabola $y^2=3x$ is parallel to the line $x+2y=1$ and the tangents at the points Q and R on the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ are…
View solution →A hyperbola passes through the foci of the ellipse ${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$ and its transverse and conjugate axes coincide with major and minor axes of…
View solution →Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is ${1 \over 2}$. If P(1, $\beta$), $\beta$ > 0 is a point on this ellipse, then the…
View solution →Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of…
View solution →The ordinates of the points P and $\mathrm{Q}$ on the parabola with focus $(3,0)$ and directrix $x=-3$ are in the ratio $3: 1$. If $\mathrm{R}(\alpha, \beta)$ is the point of…
View solution →The focus of the parabola $y^2=4 x+16$ is the centre of the circle $C$ of radius 5 . If the values of $\lambda$, for which C passes through the point of intersection of the lines…
View solution →The acute angle between the pair of tangents drawn to the ellipse $2 x^{2}+3 y^{2}=5$ from the point $(1,3)$ is :
View solution →A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola ${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1$ at the point $\left( {{x_1},{y_1}} \right)$. Then $x_1^2…
View solution →Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, ab b e( \pm 5,0)$ and $\sqrt{50}$, respectively. Then, the square of the eccentricity…
View solution →Let $\mathrm{P}$ and $\mathrm{Q}$ be any points on the curves $(x-1)^{2}+(y+1)^{2}=1$ and $y=x^{2}$, respectively. The distance between $P$ and $Q$ is minimum for some value of…
View solution →Let $x = 2t$, $y = {{{t^2}} \over 3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA \bot BA$, where A is any point on the conic. If k is…
View solution →If the point on the curve y2 = 6x, nearest to the point $\left( {3,{3 \over 2}} \right)$ is ($\alpha$, $\beta$), then 2($\alpha$ + $\beta$) is equal to _____________.
View solution →Let the equation of two diameters of a circle $x^{2}+y^{2}-2 x+2 f y+1=0$ be $2 p x-y=1$ and $2 x+p y=4 p$. Then the slope m $\in$ $(0, \infty)$ of the tangent to the hyperbola $3…
View solution →Let the normal at the point on the parabola y2 = 6x pass through the point (5, $-$8). If the tangent at P to the parabola intersects its directrix at the point Q, then the…
View solution →Let C be the largest circle centred at (2, 0) and inscribed in the ellipse ${{{x^2}} \over {36}} + {{{y^2}} \over {16}} = 1$. If (1, $\alpha$) lies on C, then 10 $\alpha^2$ is…
View solution →If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :
View solution →If the tangent to the curve, y = ex at a point (c, ec) and the normal to the parabola, y2 = 4x at the point (1, 2) intersect at the same point on the x-axis, then the value of c…
View solution →Let the eccentricity of an ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, $a b$, be ${1 \over 4}$. If this ellipse passes through the point $\left( { - 4\sqrt…
View solution →If the radius of the largest circle with centre (2,0) inscribed in the ellipse $x^2+4y^2=36$ is r, then 12r$^2$ is equal to :
View solution →If a hyperbola passes through the point P(10, 16) and it has vertices at (± 6, 0), then the equation of the normal to it at P is :
View solution →Let the tangent to the parabola $\mathrm{y}^{2}=12 \mathrm{x}$ at the point $(3, \alpha)$ be perpendicular to the line $2 x+2 y=3$. Then the square of distance of the point…
View solution →If the maximum distance of normal to the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1, b
View solution →A common tangent $\mathrm{T}$ to the curves $\mathrm{C}_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ and $C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1$ does not pass through the fourth…
View solution →Let a common tangent to the curves ${y^2} = 4x$ and ${(x - 4)^2} + {y^2} = 16$ touch the curves at the points P and Q. Then ${(PQ)^2}$ is equal to __________.
View solution →Let a line L pass through the point of intersection of the lines $b x+10 y-8=0$ and $2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right\}$. If the line $\mathrm{L}$…
View solution →If the length of the latus rectum of the ellipse $x^{2}+4 y^{2}+2 x+8 y-\lambda=0$ is 4 , and $l$ is the length of its major axis, then $\lambda+l$ is equal to ____________.
View solution →Let $P(\alpha, \beta)$ be a point on the parabola $y^2=4 x$. If $P$ also lies on the chord of the parabola $x^2=8 y$ whose mid point is $\left(1, \frac{5}{4}\right)$, then…
View solution →If m is the slope of a common tangent to the curves ${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$ and ${x^2} + {y^2} = 12$, then $12{m^2}$ is equal to :
View solution →Let an ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$, passes through $\left( {\sqrt {{3 \over 2}} ,1} \right)$ and has eccentricity ${1 \over…
View solution →Let the eccentricity of the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$ be ${5 \over 4}$. If the equation of the normal at the point $\left( {{8 \over {\sqrt…
View solution →Let the eccentricity of the hyperbola $H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$ be $\sqrt {{5 \over 2}}$ and length of its latus rectum be $6\sqrt 2$. If $y = 2x +…
View solution →Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to ___________.
View solution →Two tangent lines $l_{1}$ and $l_{2}$ are drawn from the point $(2,0)$ to the parabola $2 \mathrm{y}^{2}=-x$. If the lines $l_{1}$ and $l_{2}$ are also tangent to the circle…
View solution →An angle of intersection of the curves, ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$ and x2 + y2 = ab, a > b, is :
View solution →Let $R$ be the focus of the parabola $y^{2}=20 x$ and the line $y=m x+c$ intersect the parabola at two points $P$ and $Q$. Let the point $G(10,10)$ be the centroid of the triangle…
View solution →Let A (sec$\theta$, 2tan$\theta$) and B (sec$\phi$, 2tan$\phi$), where $\theta$ + $\phi$ = $\pi$/2, be two points on the hyperbola 2x2 $-$ y2 = 2. If ($\alpha$, $\beta$) is the…
View solution →Let the tangent and normal at the point $(3 \sqrt{3}, 1)$ on the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{4}=1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let the…
View solution →If one end of a focal chord AB of the parabola y2 = 8x is at $A\left( {{1 \over 2}, - 2} \right)$, then the equation of the tangent to it at B is :
View solution →Let the eccentricity of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is reciprocal to that of the hyperbola $2 x^{2}-2 y^{2}=1$. If the ellipse intersects the hyperbola…
View solution →Let the tangents at the points $\mathrm{P}$ and $\mathrm{Q}$ on the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$ meet at the point $R(\sqrt{2}, 2 \sqrt{2}-2)$. If $\mathrm{S}$ is…
View solution →A tangent line L is drawn at the point (2, $-$4) on the parabola y2 = 8x. If the line L is also tangent to the circle x2 + y2 = a, then 'a' is equal to ___________.
View solution →Let a 0, b 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola ${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$. Let e' and l'…
View solution →The equation of a common tangent to the parabolas $y=x^{2}$ and $y=-(x-2)^{2}$ is
View solution →If x2 + 9y2 $-$ 4x + 3 = 0, x, y $\in$ R, then x and y respectively lie in the intervals :
View solution →Let L1 be a tangent to the parabola y2 = 4(x + 1) and L2 be a tangent to the parabola y2 = 8(x + 2) such that L1 and L2 intersect at right angles. Then L1 and L2 meet on the…
View solution →Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = ____________.
View solution →If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0) a $\ne$ 0, then 'a' must be greater than :
View solution →Let $H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4 \sqrt{3}$. Suppose the point $(\alpha, 6),…
View solution →Let the parabola $y=x^2+\mathrm{p} x-3$, meet the coordinate axes at the points $\mathrm{P}, \mathrm{Q}$ and R . If the circle C with centre at $(-1,-1)$ passes through the points…
View solution →If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x+2 y=0$ is $\alpha x^2+\beta y^2-\gamma x y-30 x-60 y+225=0$, then…
View solution →Let the point P of the focal chord PQ of the parabola $y^2=16 x$ be $(1,-4)$. If the focus of the parabola divides the chord $P Q$ in the ratio $m: n, \operatorname{gcd}(m, n)=1$,…
View solution →Let $\mathrm{H}_1: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ and $\mathrm{H}_2:-\frac{x^2}{\mathrm{~A}^2}+\frac{y^2}{\mathrm{~B}^2}=1$ be two hyperbolas having length…
View solution →Let the product of the focal distances of the point $\mathbf{P}(4,2 \sqrt{3})$ on the hyperbola $\mathrm{H}: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be 32 . Let the length of the…
View solution →Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2 a$ and $2 b$, respectively, and one focus and the corresponding directrix of this…
View solution →For $0$x^2-y^2 \operatorname{cosec}^2 \theta=5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2 \operatorname{cosec}^2 \theta+y^2=5$, then the value of $\theta$ is :
View solution →Let e1 and e2 be the eccentricities of the ellipse $\frac{x^2}{b^2} + \frac{y^2}{25} = 1$ and the hyperbola $\frac{x^2}{16} - \frac{y^2}{b^2} = 1$, respectively. If b < 5 and e1e2…
View solution →Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{2}\right)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(ab)$, be $\frac{7}{4}$. Then the…
View solution →The equation of the chord, of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid-point is $(3,1)$ is :
View solution →Let the ellipse $E_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $a b$ and $E_2: \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$, $A
View solution →If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :
View solution →The length of the latus-rectum of the ellipse, whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\frac{4}{5}$, is
View solution →Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by…
View solution →Let the length of a latus rectum of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be 10. If its eccentricity is the minimum value of the function $f(t) = t^2 + t +…
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