Hyperbola Ques 20
If $2 x-y+1=0$ is a tangent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^2}{16}=1$ then which of the following CANNOT be sides of a right angled triangle?
(2017 Adv.)
(a) $a, 4,1$
(b) $2 a, 4, 1$
(c) $a, 4, 2$
(d) $2 a, 8,1$
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Answer:
Correct Answer: 20.(a, c, d)
Solution:
Formula:
Tangent $2x - y + 1 = 0$
Hyperbola $\equiv \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
It point $\equiv(a \sec \theta, b \tan \theta)$
$ \text { tangent } \equiv \frac{x \sec \theta}{a}-\frac{y \tan \theta}{b}=1 $
On comparing, we get $\sec \theta=-2 a$
$ \tan \theta=-4 \Rightarrow 4 a^{2}-16=0$
$\therefore \quad a =\frac{\sqrt{17}}{2}$
Substitute the value of $a$ in each of the options (a), (b), (c), and (d).
BETA
