Question

The focus and directrix of a parabola are $(1,2)$ and $2x-3y+1=0$. Then the equation of the tangent at the vertex is

• A. $4x-6y+5=0$
• B. $4x-6y+9=0$
• C. $4x-6y+11=0$
• D. $4x-6y+7=0$

Answer 1

The correct option is A $4x-6y+5=0$

Tangent at the vertex of parabola is parallel to its directrix.

So, the equation of directrix is $2x-3y+c=0$

This line is equidistant from the focus and the directrix.

$\therefore \left|\frac{2\left(1\right)-3\left(2\right)+c}{\sqrt{2^2+(-3{)}^2}}\right|=\left|\frac{c-1}{\sqrt{2^2+(-3{)}^2}}\right|$

$\therefore |c-4|=|c-1|$

$⟹c=\frac{5}{2}$

So, the equation of directrix is $2x-3y+\frac{5}{2}=0$

The answer is option (A).