Let P₀ is the parabola y^2=4 x with vertex K(0,0), A and B are points on P₀ where tangents drawn intersect at right angles. Let C be the centroid of A B K . The locus of C is another parabola P₁ . Now the process is repeated with P₁ then P₂, P₃ etc. Then the length of latus rectum of P₁₀ can be expressed as a b where a, b are co-prime natural numbers. Find the value of (a+ ₃ b ) .

Step-by-step solution

The answer is 4 3¹⁰ , here is the solution. Since tangents are at right angles, the line joining the point of contacts is the focal chord. Here a=1 Let A (t₁^2, 2 t₁ ) and B (t₂^2, 2 t₂ ) Since, these are end points of focal chord t₁ t₂=-1 Let t₁=t Point A (t^2, 2 t ) and B ( 1 t^2 ,- 2 t ) Let centroid C(h, k) and given K(0,0)h= t^2+ 1 t^2 +0 3 ...(1) and k= 2 t- 2 t +0 3 t- 1 t = 3 k 2 From Eqn. (1), we have aligned 3 h-2 & = (t- 1 t )^2 P₁ & =y^2= 4 3 (x- 2 3 ) aligned On repeating similar procedure gathered P₂=