Question

If the ends of a focal chord of the parabola $y^2=4ax$ are $(x_1,y_1)$ and $(x_2,y_2)$ then $x_1,x_2+y_1{y}_2=$

• A. $a^2$
• B. $-3a^2$
• C. $5a^2$
• D. $-5a^2$

Answer 1

$A=\left(x_1{y}_1\right)=(a{t}_1^2,2a{t}_1),B=(x_2,y_2)=(a{t}_1^2,2a{t}_2)$ AB is a focal chord $\Rightarrow t_1{t}_2=-1$
$\therefore x_1{x}_2+y_1{y}_2=a{t}_1^2.a{t}_2^2+2a{t}_1.2a{t}_2=a^2(t_1{t}_2{)}^2+4a^2(t_1{t}_2)=a^2(-1{)}^2+4a^2(-1)=a^2-4a^2=-3a^2$