Question

The focus directrix of a parabola are (1, 2) and x + 2y + 9 = 0 then equation of tangent at vertex is

• A. $x+2y=5$
• B. $x+2y=2$
• C. $x+2y+5=0$
• D. $x+2y+23=0$

Answer 1

The correct option is D $x+2y+23=0$

Let the distance between focus and tangent at vertex be $a$

Then the distance between focus and directrix will be $a$

Given focus is $(1,2)$ and directrix is $x+2y+9=0$

$⟹a=\frac{|1+2\left(2\right)+9|}{\sqrt{1^2+2^2}}=\frac{14}{\sqrt{5}}$

The tangent at vertex will be parallel to directrix

The equation of tangent at vertex be x+2 y+k=0

Distance between tangent at vertex and directrix is $a$

$⟹\frac{|k-9|}{\sqrt{1^2+2^2}}=\frac{14}{\sqrt{5}}$

$⟹|k-9|=14⟹k=9\pm 14=-5$ or $23$

So the equation of tangent at vertex is $x+2y+23=0$ or $x+2y-5=0$