If the sum of the squares of slopes of the normals from a point P to the hyperbola xy = c2 is equal to λ ( λ∈ R+), then the locus of the point P is–
Text Solution
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Equation of normal at any point
is ct 4 – xt 3 + ty – c = 0 .......(1)
∴ Slope of normal = t 2
The normal (1) passes through the point P (h, k)
∴ ct 4 – ht 3 + kt – c = 0. If the roots of this equation are t i ; i = 1, 2, 3, 4
then Σ t i = 
and Σ t i t j = 0,
Σ t i t j t k = – 
and t 1 t 2 t 3 t 4 = – 1
Given Σ t i2 = λ ⇒ ( Σ t i ) 2 – 2 Σ t i t j = λ
⇒ 
– 0 = λ ⇒ h 2 = λ c 2
∴ Required locus is x 2 = λ c 2
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