Let tangents drawn from point C(0,-b) to hyperbola frac{{{x^2}}}{{{a^2}}}

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10-2. Parabola, Ellipse, Hyperbola

normal

Let tangents drawn from point $C(0,-b)$ to hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ touches hyperbola at points $A$ and $B.$ If $\Delta ABC$ is a right angled triangle, then $\frac{a^2}{b^2}$ is equal to -

$1$

$\frac{1}{2}$

$2$

$\frac{3}{2}$

Let $\mathrm{A}(\mathrm{a} \sec \theta, \mathrm{btan} \theta)$

tangent at $A$ is $\frac{x \sec \theta}{a}-\frac{y \tan \theta}{b}=1$

$\because$ tangent passes through $(0,-b) \Rightarrow \tan \theta=1$

$\therefore A(a \sqrt{2}, b)$

$\Rightarrow \mathrm{CA} \cos 45^{\circ}=\mathrm{CM}$

$\Rightarrow \sqrt{2 a^{2}+4 b^{2}} \cdot \frac{1}{\sqrt{2}}=2 b$

$\Rightarrow \mathrm{a}^{2}=2 \mathrm{b}^{2}$

Standard 11

Mathematics

easy

medium

hard

(IIT-2007)

medium

normal