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10-2. Parabola, Ellipse, Hyperbola
hard
If the point of intersections of the ellipse $\frac{ x ^{2}}{16}+\frac{ y ^{2}}{ b ^{2}}=1$ and the circle $x ^{2}+ y ^{2}=4 b , b > 4$ lie on the curve $y^{2}=3 x^{2},$ then $b$ is equal to:
A
$12$
B
$5$
C
$6$
D
$10$
(JEE MAIN-2021)
Solution
$y^{2}=3 x^{2}$
And $x^{2}+y^{2}=4 b$
Solve both we get
So $x^{2}=b$
$\frac{x^{2}}{16}+\frac{3 x^{2}}{b^{2}}=1$
$\frac{b}{16}+\frac{3}{b}=1$
$b^{2}-16 b+48=0$
$(b-12)(b-4)=0$
$b=12, b > 4$
Standard 11
Mathematics