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
4x6y+5=0
B
4x6y+9=0
C
4x6y+11=0
D
4x6y+7=0

Solution

## 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.$$\therefore \left | \dfrac{2(1)-3(2)+c}{\sqrt{2^2+(-3)^2}} \right |=\left |\dfrac{c-1}{\sqrt{2^2+(-3)^2}} \right |$$$$\therefore \left | c-4 \right |=\left | c-1 \right |$$$$\implies c=\dfrac{5}{2}$$So, the equation of directrix is $$2x-3y+\dfrac{5}{2}=0$$The answer is option (A).Mathematics

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