Two mutually perpendicular tangents of the parabola y2 = 4ax meet the axis in P1 and P2. If S is the focus of the parabola then \frac{1}{ℓ \left(\right. \left(SP\right)_{1} \left.\right)} + \frac{1}{ℓ \left(\right. \left(SP\right)

Engineering

Mathematics

Tangent of Parabola

Question

Two mutually perpendicular tangents of the parabola y² = 4ax meet the axis in P₁ and P₂. If S is the focus of the parabola then $\frac{1}{ℓ ({SP}_{1})} + \frac{1}{ℓ ({SP}_{2})}$ is equal to

$\frac{4}{a}$

$\frac{1}{4 a}$

$\frac{2}{a}$

$\frac{1}{a}$

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SP1 = a(1 + $t_{1}^{2}$ ) ; SP2 = a(1 + $t_{2}^{2}$ )

also t₁t₂ = – 1

$\frac{1}{{SP}_{1}}$ = $\frac{1}{a (1 + t^{2})}$ ; $\frac{1}{{SP}_{2}}$ = $\frac{t^{2}}{a (1 + t^{2})}$

∴ $\frac{1}{{SP}_{1}}$ + $\frac{1}{{SP}_{2}}$ = $\frac{1}{a}$

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