Question

Assertion :Director circle of hyperbola $\frac{x^2}{a^2}\frac{y^2}{b^2}+1=0$ is defined only when $b>a$. Reason: Director circle of hyperbola $\frac{x^2}{25}\frac{y^2}{9}=1$ is $x^2+y^2=16.$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Since the director circle is the locus of point of intersection
of perpendicular tangents any tangent in terms of $m$ of $\frac{x^2}{a^2}-\frac{y^2}{b^2}+1=0$ is

$y=mx(b^2-a^2{m}^2)$

or $(ymx{)}^2=b^2{a}^2{m}^2$

so, $m^2(x^2+a^2)-2mxy+y^2-b^2=0$

since, $m_1{m}_2=-1$

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

or $x^2+y^2=b^2-a^2,b>a$

Also, director circle of $\frac{x^2}{25}-\frac{y^2}{9}=1$ is

$x^2+y^2=25-9=16.$

So,

the correct answer is B.