the radius of the circle passing through the foci of the

The radius of the circle passing through the foci of the ellipse $$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1,$$ and having its centre at $$\left( {0,\,3} \right)$$ is-

A. $$4$$

B. $$3$$

C. $$\sqrt {\frac{1}{2}} $$

D. $$\frac{7}{2}$$

Answer : $$4$$

Solution :
For ellipse $$\frac{{{x^2}}}{{{4^2}}} + \frac{{{y^2}}}{{{3^2}}} = 1,\,\,a = 4,\,\,b = 3$$
$$ \Rightarrow e = \sqrt {1 - {{\left( {\frac{3}{4}} \right)}^2}} = \frac{{\sqrt 7 }}{4}$$
$$\therefore $$ Foci are $$\left( {\sqrt 7 ,0} \right)$$ and $$\left( { - \sqrt 7 ,0} \right)$$
Centre of circle is at (0, 3) and it passes through $$\left( { + \sqrt 7 ,0} \right),$$ therefore radius of circle $$ = \sqrt {{{\left( {\sqrt 7 } \right)}^2} + {{\left( 3 \right)}^2}} = 4$$

Releted MCQ Question on
Geometry >> Ellipse

Releted Question 1

Let $$E$$ be the ellipse $$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$$ and $$C$$ be the circle $${x^2} + {y^2} = 9.$$ Let $$P$$ and $$Q$$ be the points $$\left( {1,\,2} \right)$$ and $$\left( {2,\,1} \right)$$ respectively. Then-

A. $$Q$$ lies inside $$C$$ but outside $$E$$

B. $$Q$$ lies outside both $$C$$ and $$E$$

C. $$P$$ lies inside both $$C$$ and $$E$$

D. $$P$$ lies inside $$C$$ but outside $$E$$

View Answer

Releted Question 2