If $(2,3)$ is the focus and $x-y+3=0$ is the directrix of a parabola then the equation of the tangent drawn at the vertex of the parabola is

• A

  $x-y-2=0$
• Correct answer
  B

  $x-y+2=0$
• C

  $x-y+5=0$
• D

  $x-y-5=0$

Solution

Foot of perpendicular from focus on directrix be $(h, k)$ $\begin{aligned} & \Rightarrow \frac{h-2}{1}=\frac{k-3}{-1}=\frac{-(2+3-3)}{2} \\ & \Rightarrow h-2=\frac{k-3}{-1}=-1 \Rightarrow h=1, k=4 \\ & \text { Vertex }=\left(\frac{1+2}{2}, \frac{4+3}{2}\right)=\left(\frac{3}{2}, \frac{7}{2}\right) \end{aligned}$
Equation of tangent at vertex $=\frac{y-\frac{7}{2}}{x-\frac{3}{2}}=1$ $\Rightarrow y=x+2$.

Hence, the correct answer is option: 'B'.