Question #224001

The coordinates of the ends of a focal chord of the parabola y2= 4ax are (x1, y1) and (x2, y2). Show that x1x2 = a2 and y1y2 = −4a2.


1
Expert's answer
2021-08-19T15:12:14-0400

Answer:-

Let P(x1,y1)=(at12,2at1)P(x_1,y_1) = (at_1^2 , 2at_1) and Q(x2,y2)=(at22,2at2)Q(x_2, y_2) = (at_2^2 , 2at_2) be two end points of a focal chord. P, S, Q are collinear.

Slope of PS = Slope of QS

2at1at12a=2at2at22at1t22t1=t2t12t2t1t2(t2t1)+(t2t1)=0t1t2=1\dfrac{2at_1}{at_1^2-a}=\dfrac{2at_2}{at_2^2-a}\\ t_1t_2^2-t_1=t_2t_1^2-t_2\\ t_1t_2(t_2-t_1)+(t_2-t_1)=0\\ t_1t_2=-1

from (1)

x1x2=at12at22=a2(t1t2)2=a2y1y2=2at12at2=4a2(t1t2)=4a2x_1x_2=at_1^2at_2^2=a^2(t_1t_2)^2=a^2\\ y_1y_2=2at_12at_2=4a^2(t_1t_2)=-4a^2\\





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