A pair of tangents to a parabola is are equally inclined to a straight line whose inclination to the axis is α. The locus of their point of intersection is :

Engineering

Mathematics

Tangent of Parabola

Question

a circle

a parabola

a straight line

a line pair

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Let P $({at}_{1} t_{2}, a (t_{1} + t_{2}))$ = P(h, k)
slope of tangents at A, m₁ = $\frac{1}{t_{1}}$ and at B, m₂ = $\frac{1}{t_{2}}$
Let m = tan α
then $\frac{m_{1} - m}{1 + {mm}_{1}} = \frac{m - m_{2}}{1 + {mm}_{2}}$

       tan(θ₁ – α) = tan(α – θ₂)
∴ (θ₁ – α) = nπ + 2aα
∴   tan 2α = tan (θ₁ + θ₂) = $\frac{{tanθ}_{1} + {tanθ}_{2}}{1 - {tanθ}_{1} {tanθ}_{2}}$ = $\frac{\frac{1}{t_{1}} + \frac{1}{t_{2}}}{1 - \frac{1}{t_{1}} \frac{1}{t_{2}}} = \frac{2 (t_{1} + t_{2})}{t_{1} t_{2} - 1}$ ; tan 2α = $\frac{\frac{k}{a}}{\frac{h}{a} - 1} = \frac{k}{h - a}$
∴   locus of P(h, k) will be y = (x – a)tan 2α

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