Question

Assertion :The equation of the director circle to the ellipse $4x^2+25y^2=100$ is $x^2+y^2=29$. Reason: Director circle is the locus of point of intersection of $\perp$ er tangents to an ellipse.

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Any tangent in terms of slope of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $y=mx\pm \sqrt{a^2{m}^2+b^2}\Rightarrow (y-mx{)}^2=a^2{m}^2+b^2$
$\Rightarrow (x^2-a^2)m^2-2mxy+y^2-b^2=0.....(\ast )$ which is quadratic in $m$. Let $m_1,m_2$ are roots of $(\ast )$

$\therefore m_1{m}_2=-1\Rightarrow \frac{y^2-b^2}{x^2-a^2}=-1$
$\Rightarrow x^2+y^2=a^2+b^2$ is director circle of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
The equation of the director circle of $\frac{x^2}{25}+\frac{y^2}{4}=1$ is $x^2+y^2=25+4=29$