Question

If the ellipse $4x^2+9y^2=36$ and the hyperbola $a^2{x}^2-y^2=4$ intersect at right angles, then which among the following options are correct

• A. $a=2$
• B. $a=-4$
• C. Equation of circle through the points of intersection of two conic is $x^2+y^2=5$
• D. Equation of circle through the points of intersection of two conic is $x^2+y^2=25$

Answer 1

Answer: (A), (C)

The correct options are
A $a=2$
C Equation of circle through the points of intersection of two conic is $x^2+y^2=5$

If ellipse and hyperbola are having same centre and intersecting each other orthogonally, then thay will be confocal
$\therefore \frac{2}{a}e=\sqrt{5}\Rightarrow \frac{2}{a}\sqrt{1+\frac{4}{4/a^2}}=\sqrt{5}\Rightarrow a=\pm 2$
Now equation of the curve which passes through the intersection of
$E:4x^2+9y^2=36$ and
$H:4x^2-y^2=4$ is
$E+\lambda H=0\Rightarrow 4x^2(1+\lambda )+y^2(9-\lambda )-36-4\lambda =0$
For the curve to represent the circle
$4(1+\lambda )=9-\lambda \Rightarrow \lambda =1$
Hence equation of the required circle
$x^2+y^2=5$