"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 :"

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Accepted Answer

"3x + 4y = 12$\sqrt 2$

$\Rightarrow$ y = $- {{3x} \over 4} + 3\sqrt 2$ is tangent to

${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1$

$\therefore$ c² = a²m² + b²

$\Rightarrow$ ${\left( {3\sqrt 2 } \right)^2} = {a^2}{\left( { - {3 \over 4}} \right)^2} + 9$

$\Rightarrow$ a² = 16

Also e = $\sqrt {1 - {{{b^2}} \over {{a^2}}}}$

= $\sqrt {1 - {9 \over {16}}}$ = ${{\sqrt 7 } \over 4}$

Distance between focii = 2ae

= $2 \times 4 \times {{\sqrt 7 } \over 4}$

= $2\sqrt 7$"

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 :

Solution

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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 :

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More Conic Sections questions

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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 :