Two straight lines are perpendicular to each other. One of them touches the parabola  y2=4ax+a and the other touches  y2=4bx+b .Their point of intersection lies on the line

Given parabola is  y2=4ax+aSlope form of equation of tangent is  y=mx+a+am…..(i) Any tangent to y2=4b(x+b) which is perpendicular to (i) is y=−1m(x+b)−bm… (ii) Subtracting (i) & (ii),  we get m+1mx+(a+b)m+1m=0⇒x+a+b=0Which is a locus of their point of intersection.