Question

If the director circle of ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of eccentricity $\frac{1}{\sqrt{3}}$ passes through the two foci of the hyperbola $H:\frac{y^2}{9}-\frac{x^2}{16}=1$ then

• A. equation of director circle of ellipse is $x^2+y^2=25$.
• B. Area of quadrilateral formed by tangents at ends of latus rectum of ellipse $E$ is $30\sqrt{3}$ sq. units.
• C. Area of square formed by four tangents to ellipse $E$ is 50 sq. units
• D. number of points on ellipse from which two perpendicular tangents can be drawn to hyperbola $H$ is 4 .

Answer 1

Answer: (A), (B), (C)

$e_H^2=1+\frac{16}{9}=\frac{25}{9}\Rightarrow e_H=\frac{5}{3}\Rightarrow \text{ foci }(0,\pm 5)$
$\therefore a^2+b^2=25\text{ and }{a}^2-b^2=\frac{a^2}{3}\Rightarrow \frac{2a^2}{3}=b^2$
$\frac{5a^2}{3}=25\Rightarrow a^2=15\text{ and }{b}^2=10$
$\therefore E:\frac{x^2}{15}+\frac{y^2}{10}=1\Rightarrow A=\frac{2a^2}{e}=\frac{2\cdot 15}{\frac{1}{\sqrt{3}}}=30\sqrt{3}$
Area of square formed by tangents $={\left(\frac{2R_{\text{Director }}}{\sqrt{2}}\right)}^2=\frac{4\cdot 25}{2}=50$ sq. units