Question

Consider a parabola with focus $(1,0)$ whose tangent at $(4,2)$ is $y=x-2.$ Then which of the following is (are) CORRECT ?

• A. Equation of axis of symmetry is $3x-2y=3$
• B. Length of latus rectum is $\frac{2}{\sqrt{13}}$ unit
• C. Equation of directrix is $2x+3y=1$
• D. Equation of parabola is $9x^2+4y^2-12xy-22x+6y+12=0$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Equation of axis of symmetry is $3x-2y=3$
B Length of latus rectum is $\frac{2}{\sqrt{13}}$ unit
C Equation of directrix is $2x+3y=1$
D Equation of parabola is $9x^2+4y^2-12xy-22x+6y+12=0$

Mirror image $B$ of focus $F$ about the tangent $y=x-2$ lies on the directrix.
Also, $PB$ is parallel to axis of parabola.


$\frac{x-1}{1}=\frac{y-0}{-1}=\frac{-2(1-0-2)}{1^2+(-1{)}^2}$
$\Rightarrow \frac{x-1}{1}=\frac{y}{-1}=1$
$\Rightarrow x=2$ and $y=-1$

Line $PB:(y+1)=\frac{3}{2}(x-2)$
$\Rightarrow 3x-2y=8$
Equation of axis is $3x-2y=\lambda ,$ satisfying $(1,0).$
$\Rightarrow$ Equation of axis of symmetry is $3x-2y=3$

Equation of directrix is $2x+3y=m$, satisfying $(2,-1)$
$\Rightarrow$ Equation of directrix is $2x+3y=1$

Length of semi-latus rectum $=\frac{|2+0-1|}{\sqrt{2^2+3^2}}=\frac{1}{\sqrt{13}}$

Also, distance of a point from focus is equal to the distance of point from the directrix.
$\Rightarrow \sqrt{(x-1{)}^2+y^2}=\frac{|2x+3y-1|}{\sqrt{2^2+3^2}}$
$\Rightarrow 13(x^2-2x+1+y^2)=4x^2+9y^2+1+12xy-4x-6y$
$\Rightarrow 9x^2+4y^2-12xy-22x+6y+12=0$