Question

The ellipse 4x² + 9y² = 36 and the hyperbola a²x² – y² = 4 intersect at right angles then the equation of the circle through the points of intersection of two conic is

(a)  x² + y² = 25 

(b)  5(x² + y²) + 3x + 4y = 0

(c)  5(x² + y²) – 3x – 4y = 0

(d)  (x² + y²) = 5 

Answer 1

Correct option  (d) (x² + y²) = 5

Explanation:

Since ellipse and hyperbola intersect orthogonally, they are confocal.

e = √1 - 4/9 = √5/3

foci of ellipse (±√5, 0)

(ae)² =  a² + b

5 = 4/a² + 4 

a = 2

Let point of intersection in the first quadrant be P(x₁² + 9y²₁ = 4, y1 ).

P lies on both the curves.

4x₁² + 9y₁² = 36 and 4x₁² - y₁² = 4

Adding these two, we get

8x₁² + 8y₁² = 40

x₁² + y₁² = 5

Equation of circle is x² + y² = 5