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Question

Let normal's drawn to parabola at point's P(0,0) and Q(3,−1) intersect at (2,1). If PQ is bisected by the axis of the parabola, then

A
Equation of directrix is x+3y+5=0
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B
Slope of axis is 3
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C
Focus is (8,0)
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D
Slope of tangent at vertex is 13
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Solution

The correct options are
A Equation of directrix is x+3y+5=0
B Slope of axis is 3
Clearly normals are perpendicular to each other.
So, quadrilateral formed by tangents and normals at given points here forms a rectangle.
axis of the parabola bisects the PQ and tangents drawn to the ends of the chord are perpendicular
PQ is the latusrectum of the given parabola whose focus is (32,12).
Hence tangents will intersect at (1,2)

directrix is parallel to latusrectum
Slope of directrix =slope of tangent at vertex=13 and
Slope of axis =3
So, equation of directrix is y+2=13(x1)
x+3y+5=0

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