Paragraph: Consider the following curves. aligned & C₁: y^2=4 x & C₂: y^2(a+b x)=(a+c x)^2, a, b, c I⁺ aligned where C₂ is the locus of the family of circles which passes through focus of C₁ and cuts C₁ orthogonally. On the basis of above information, answer the following questions: Question: The area bounded by the curve C₂ & the ordinates x=- 1 3 and x=1 -

Step-by-step solution

Let the circle be x^2+y^2+2 g x+2 f y+c=0 If passes through (1,0) 1+2 g+c=0 ...(i) Let (t^2, 2 t ) be the point of intersection of two curves equation of tangent is yt = x + t ^2 ...(ii) Circle & parabola cuts orthogonaly (ii) passes through the centre of the circle -f t=t^2-gf t=g-t^2 ...(iii) ( t ^2, 2 t ) also satisfies equation of circle t^4+4 t^2+2 gt ^2+4 f t+c=0 Putting the value of c & f t in above equation aligned & t ^4+4 t ^2+2 gt ^2+4 ~g -4 t ^2-1-2 ~g =0 & t ^4+2 gt ^2+2 ~g -1=0 & ( t ^4-1 )+2 ~g ( t ^