Q 1 Prove that that the length of the chord joining the points of contact of tangents drawn - Maths - Application of Derivatives

Question

Q 1 Prove that that the length of the chord joining the points of
contact of tangents drawn from the point x 1 ,   y 1   i
s   y 1 2 + 4 a 2 y 1 2 - 4 a x 1 a

Aarushi Mishra · Accepted Answer

Comsider parabolay2=4axLet point of contact of tangents drawn from x1,y1 be At1 and B t2A=at12,2at1B=at22,2at2AB=at12-at222+2at1-2at22AB=a2t12-t22+4a2t1-t22AB=a2t1-t2t1+t22+4a2t1-t22AB=a2t1-t22t1+t22+4AB=at1-t2t1+t22+4AB=at1-t22t1+t22+4AB=at12+t22-2t1t2t1+t22+4AB=at12+t22+2t1t2-4t1t2t1+t22+4AB=at1+t22-4t1t2t1+t22+4We know that point of intersection tangent at At1 and Bt2 is given by at1t2, at1+t2x1,y1=at1t2, at1+t2x1=at1t2t1t2=x1ay1=at1+t2t1+t2=y1aAB=ay1a2-4x1ay1a2+4AB=ay12-4ax1a2y12+4a2a2AB=y12-4ax1y12+4a2a

Q.1. Prove that that the length of the chord joining the points of contact of tangents drawn from the point x 1 , y 1 i s y 1 2 + 4 a 2 y 1 2 - 4 a x 1 a

Q.1. Prove that that the length of the chord joining the points of contact of tangents drawn from the point $(x_{1}, y_{1}) i s \frac{\sqrt{{y_{1}}^{2} + 4 a^{2}} \sqrt{{y_{1}}^{2} - 4 a x_{1}}}{a}$