A line which makes an acute angle $θ$ with the positive direction of x-axis is drawn through the point P (3,4) and cuts the curve $y^{2} = 4 x$ at Q&R the lengths of segments PQ and PR are numerical values of roots of equation

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$r^{2} \sin^{2} θ + 4 r (2 \sin θ - \cos θ) + 4 = 0$

$r^{2} \sin^{2} θ + 4 r (2 \sin θ + \cos θ) + 4 = 0$

$r^{2} \sin^{2} θ - 4 r (2 \sin θ + \cos θ) + 4 = 0$

$r^{2} \sin^{2} θ - 4 r (2 \sin θ - \cos θ) + 4 = 0$

answer is B.

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Detailed Solution

A point on line through P(3,4) and having inclination $θ$ is $Q (3 + r \cos θ, 4 + r \sin θ)$

$Q l i e s a n y^{2} = 4 x$ ${(4 + r \sin θ)}^{2} = 4 (3 + r \cos θ)$

$\Rightarrow r^{2} \sin^{2} θ + 4 r (2 \sin θ - \cos θ) + 4 = 0$

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