Question

The equation of the circle whose end points of the diameter are the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ is

• A. $x^2+y^2=16$
• B. $x^2+y^2=23$
• C. $x^2+y^2=7$
• D. $x^2+y^2=9$

Answer 1

The correct option is D $x^2+y^2=9$

Given ellipse: $\frac{x^2}{16}+\frac{y^2}{7}=1$
Comparing this with
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
We get
$a=4,b=\sqrt{7}e=\sqrt{1-\frac{b^2}{a^2}}\Rightarrow e=\sqrt{1-\frac{7}{16}}\Rightarrow e=\frac{3}{4}$

Foci $=(\pm ae,0)=(\pm 3,0)$
Now, the equation of the circle in diametric form is
$(x+3)(x-3)+(y-0)(y-0)=0\therefore x^2+y^2=9$