Question

The focus and directrix of a parabola are \( (1,2) \) and \( x+2 y+9=0 \) then equation of tangent at vertex is 1) \( x+2 y=5 \) 2) \( x+2 y=2 \) 3) \( x+2 y+5=0 \) 4) \( x+2 y+2=0 \)

Answer 1

The correct Option is (1)

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=|1+2(2)+9|√12+22=14√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

⟹|k−9|√12+22=14√5

⟹|k−9|=14⟹k=9±14=−5 or 23

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