Question

If $(x_1,y_1)$ and $(x_2,y_2)$ are the extremeties of the focal chord of parabola $y^2=16ax$, then the value of $16x_1{x}_2+y_1{y}_2$ is

Answer 1

We know that,

For a Parabole $y^2=4ax$

In parametric form,

If one end of the focal chord of the parabola is $(a{t}_1^2,4a{t}_1)$ then,

Other end of the focal chord of the parabola is $(a{t}_2^2,4a{t}_2)$.

i.e., $(x_1,y_1)=(a{t}_1^2,4a{t}_1)$ and

$(x_2,y_2)=(a{t}_2^2,4a{t}_2)$
For extremities of focal chord of the parabola we know that $t_1{t}_2=-1$.
$16x_1{x}_2+y_1{y}_2$

$=16a{t}_1^{2}a{t}_2^2+4a{t}_{1}4a{t}_2$

$=16a^2{t}_1^2{t}_2^2+16a^2{t}_1{t}_2$

$=16(t_1{t}_2{)}^2+16t_1{t}_2$

$=16(-1{)}^2+16(-1)$

$=16-16$

$=0$