If a chord joining P a Sec - a tan- Q a Sec - a tan- on the hyperbola x 2- y 2- a 2 is the normal at P-then T an -A- Tan-4 2-1B- Tan-4 2-1- Tan-2 2-1D- Tan-1-2 Sec2-

The correct option is B Slope of chord joining P and Q = slope of normal at P Tan α Tan θ/sec α sec θ=Tan θ/sec θ⇒ Tan α Tan θ = k Tan θ and sec α sec θ ...

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Standard XII

Mathematics

Directrix of Hyperbola

If a chord jo...

Question

If a chord joining $P (a S e c θ, a t a n θ), Q (a S e c α, a t a n α)$ on the hyperbola $x^{2} - y^{2} = a^{2}$ is the normal at P,then $T a n α$

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Solution

The correct option is B

Slope of chord joining P and Q = slope of normal at P

$\frac{T a n α - T a n θ}{s e c α - s e c θ} = \frac{T a n θ}{s e c θ} \Rightarrow T a n α - T a n θ = - k T a n θ a n d s e c α - s e c θ = k s e c θ$

$∴ (1 - k) T a n θ = T a n α \to 1. (1 + k) s e c θ = s e c α \to 2$

${[(1 + k) s e c θ]}^{2} - {[(1 - k) T a n θ]}^{2} = s e c^{2} α - T a n^{2} α$

$\Rightarrow k = - 2 (s e c^{2} θ + T a n^{2} θ) = - 4 s e c^{2} θ + 2$

$F r o m (1) T a n α = T a n θ (1 + 4 s e c^{2} θ - 2) = T a n θ (4 s e c^{2} θ - 1)$

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If a chord joining P(a Sec θ, a tan θ), Q(a Sec α,a tan α) on the hyperbola x2−y2=a2 is the normal at P,then T an α

If a chord joining $P (a S e c θ, a t a n θ), Q (a S e c α, a t a n α)$ on the hyperbola $x^{2} - y^{2} = a^{2}$ is the normal at P,then $T a n α$