Answer to Question #224001 in Analytic Geometry for Bless

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(x_1,y_1) = (at_1^2 , 2at_1)" and "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

"\\dfrac{2at_1}{at_1^2-a}=\\dfrac{2at_2}{at_2^2-a}\\\\\nt_1t_2^2-t_1=t_2t_1^2-t_2\\\\\nt_1t_2(t_2-t_1)+(t_2-t_1)=0\\\\\nt_1t_2=-1"

from (1)

"x_1x_2=at_1^2at_2^2=a^2(t_1t_2)^2=a^2\\\\\ny_1y_2=2at_12at_2=4a^2(t_1t_2)=-4a^2\\\\"





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