49 A circle with centre Q having double contact with parabol

49, A circle with centre Q having double contact with parabola y?-(2+/3)x passes through its focus S. If tangent at their point of contact P intersect its axis at R, then radius of inscribed circle of triangle PSR is 1/3 4 /3 D) B) C) V3

Solution

Consider parabola ---

y² = (2+3)x

For a general parabola, y² = 4ax

Focus - (a,0)

Here, 4a = 2+3

a = (2+3)/4

S = {(2+3)/4,0}

S = (1/2 + 3/4 , 0)

Now circle passes through S.

So tangent at P will intersect axis at R.

So triangle PSR will be right angle triangle.

So center of inscribed circle will be at half of hypotenuse.

So, radius = 1/2 + 3/4 - 1/2 = 3/4