Question

Let normals drawn to parabola at points $P(0,0)$ and $Q(3,-1)$ intersect at $(2,1)$. If $PQ$ is bisected by the axis of the parabola, then

• A. Equation of directrix is $x+3y+5=0$
• B. Slope of axis is $3$
• C. Focus is $(8,0)$
• D. Slope of tangent at vertex is $\frac{1}{3}$

Answer 1

Answer: (A), (B)

Slope of axis is $3$

Clearly normals are perpendicular to each other.
So, quadrilateral formed by tangents and normals at given points here forms a rectangle.
$\because$ axis of the parabola bisects the $PQ$ and tangents drawn to the ends of the chord are perpendicular,
$\therefore PQ$ is the latus rectum of the given parabola whose focus is $(\frac{3}{2},\frac{-1}{2})$.
Hence tangents will intersect at $(1,-2)$

$\because$ directrix is parallel to latus rectum,
$\therefore$ Slope of directrix $=$ Slope of tangent at vertex $=-\frac{1}{3}$ and
Slope of axis $=3$
So, equation of directrix is $y+2=-\frac{1}{3}(x-1)$
$\Rightarrow x+3y+5=0$