Question

Two equal circles of radius $\sqrt[2]{5}$ passes through the entries LR of $[tex]y^{2}$ = 4x[/tex] then find the dist. between centries of circles

Answer 1

Let the latus ractum of the parabola y² = 4x be the common chord to the circles C₁ and C₂ each of them having radius 2√5. Then, the distance between the centres of the circles C₁ and C₂ is ...

solution : latus rectum of parabola is chord of circles. so, LL' perpendicular on C₁C₂.

let LL' cuts C₁C₂ at M.

then LM = L'M = LL'/2 and C₁C₂ = 2 C₁M

length of latus rectum = 4a = 4 × 1 = 4

so, LL' = 4

so, LM = LL'/2 = 4/2 = 2

given, C₁L = radius of circle = 2√5

now using Pythagoras theorem,

C₁M² = C₁L² - LM²

= (2√5)² - (2)² = 20 - 4 = 16

so, C₁M = 4 ⇒C₁C₂ = 2 × 4 = 8

Therefore the distance between centre of circles is 8