The number of integral values of k for which
the line, 3x + 4y = k intersects the circle,
x2
+ y2
– 2x – 4y + 4 = 0 at two distinct points is
______.
Answer (integer)
9
Solution
Circle x<sup>2</sup>
+ y<sup>2</sup>
– 2x – 4y + 4 = 0
<br><br>$\Rightarrow$ (x – 1)<sup>2</sup>
+ (y – 2)<sup>2</sup>
= 1
<br><br>Centre: (1, 2), radius = 1
<br><br>Line 3x + 4y – k = 0 intersects the circle at two distinct points.
<br><br>$\Rightarrow$ distance of centre from the line < radius
<br><br>$\Rightarrow$ $$\left| {{{3 \times 1 + 4 \times 2 - k} \over {\sqrt {{3^2} + {4^2}} }}} \right| < 1$$
<br><br>$\Rightarrow$ |11 - k| < 5
<br><br>$\Rightarrow$ 6 < k < 5
<br><br>$\Rightarrow$ k $\in$ {7, 8, 9, ……15} since k $\in$ I
<br><br>$\therefore$ Total 9 integral value of k.
About this question
Subject: Mathematics · Chapter: Circles · Topic: Equation of a Circle
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