Question

An ellipse $\frac{x^z}{4}+\frac{y^z}{3}=1$ confocal with hyperbola $\frac{x^2}{{\cos }^2\theta }-\frac{y^2}{{\sin }^2\theta }=1$ then the set of value of ${}^{\prime }{0}^{\prime }$

• A. $R$
• B. $R-\{n\pi ,nϵz\}$
• C. $R-\{(2n+1)\frac{\pi }{2},nϵz\}$
• D. $R-\{\frac{n\pi }{2},nϵz\}$

Answer 1

Answer: (A)

$R$

Focus of ellipse$=ae=a\sqrt{1-\frac{b^2}{a^2}}$

$=\sqrt{a^2-b^2}$

$=\sqrt{1}$

$=1$

Focus of hyperbola$=a\sqrt{1+\frac{b^2}{a^2}}$

$=\sqrt{a^2+b^2}$

$=\sqrt{{\sin }^2\theta +{\cos }^2\theta }$

$=\sqrt{1}$

$=1$

$\therefore$The ellipse and hyperbola will be confocal for $\theta ϵR$.