"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?"

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Accepted Answer

"${{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1$

$\therefore$ c = $\pm$ $\sqrt {{a^2}{m^2} - {b^2}}$

$\Rightarrow$ c = $\pm$ $\sqrt {100{m^2} - 64}$

General tangent to hyperbola in slope form is

y = mx $\pm$ $\sqrt {100{m^2} - 64}$

This tangent is also tangent to the circle x² + y² = 36, whose center (0, 0) and radius = 6.

Distance of the tangent from the center is

$\left| {{{\sqrt {100{m^2} - 64} } \over {\sqrt {{m^2} + 1} }}} \right|$ = 6

$\Rightarrow$ 100m² — 64 = 36m² + 36

$\Rightarrow$ 64m² = 100

$\Rightarrow$ m = ${{10} \over 8}$

$\therefore$ c² = 100 $\times$ ${{100} \over {64}}$ - 64

$\Rightarrow$ c² = ${{{{100}^2} - {{64}^2}} \over {64}}$

$\Rightarrow$ c² = ${{164 \times 36} \over {64}}$

$\Rightarrow$ 4c² = 369"

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?

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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?

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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?