Question

$P_1:y^2=4ax,P_2:y^2=-4axL:y=x$

The coordinates of other extremity of a focal chord of the parabola $P_2$, one of whose extremity is the point of intersection of $L$ and $P_2$ is

• A. $(-a,2a)$
• B. $(-\frac{a}{4},a)$
• C. $(-\frac{a}{4}-a)$
• D. $(-a,-2a)$

Answer 1

The correct option is B $(-\frac{a}{4},a)$

$P_1(y^2=4ax)$
$P_2(y^2=-4ax)$
$L(y=x)$
Focus of Parabola $P_2$ is $F_2(-a,0)$
Line $L$ and Parabola $P_2$ intersects at point $A(-4a,-4a)$, which is also a extremity of a focal chord of $P_2$.
Other extremity of Focal chord can be obtained through a Parabolic property $t_1{t}_2=-1$,
$\Rightarrow t_1=-2,t_2=\frac{1}{2}$
$\Rightarrow$ Extremities of Focal chord of $P_2$ are $A(-4a,-4a)$ and $B(\frac{-a}{4},a)$.
Hence, Option B