Question

The tangent at the vertex of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets its conjugate hyperbola at the point whose coordinates are

• A. $(a,\sqrt{2}b)$
• B. $(\sqrt{2}a,b)$
• C. $(-\sqrt{2}a,b)$
• D. $noneofthese$

Answer 1

Answer: (A)

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

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.....(1)

conjugate hyperbola :

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$.....(2)

vertex$=(\pm a,0)$ tangent vertex $=\pm \frac{x}{a}=1$

$\Rightarrow x=\pm a$

substituting in (2),

$2=\frac{y}{b^2}$

$\Rightarrow$ points are $(\pm a,\pm \sqrt{2}b)$

among the options given,$A=(a,\sqrt{2}b)$ is the right answer.