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