Question

If the ends of a focal chord of the parabola $y^2=8x$ are $(x_1,y_1)$ and $(x_2,y_2),$ then $x_1{x}_2+y_1{y}_2$ is equal to :

• A. $12$
• B. $10$
• C. $0$
• D. $-12$
• E. $20$

Answer 1

Answer: (D)

$-12$

Given equation of parabola is $y^2=8x$
Therefore, $a=2$
Let end points of a focal chord are
$(2t_1^2,4t_1)$ and $(2t_2^2,4t_2)$
Let $(x_1,y_1)=(2t_1^2,4t_1)$
and $(x_2,y_2)=(2t_2^2,4t_2)$
Thus $x_1{x}_2+y_1{y}_2=2t_1^2\times 2t_2^2+4t_1\times 4t_2$
$=4(t_1{t}_2{)}^2+16(t_1{t}_2)$
$=4(-1{)}^2+16(-1)$ ....$(\text{Since}{t}_1{t}_2=-1)$
$=4-16$
$=-12$