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if normal on hyperbola at any point on it meet it again then

if normal on hyperbola at any point on it meet it again then

Grade:12th pass

1 Answers

Sumit Majumdar IIT Delhi
askIITians Faculty 137 Points
9 years ago
Dear student,

We know that the equation of normal for a hyperbola would be given by:
∵ xy = c²
Hence, y = c² (1/x)
so, we get
dy/dx = -c²/x²

Thus at any point P(t₁) ≡ ( ct₁, c/t₁ ), we would have the slope of normal as given by:

m = -1 / (dy/dx ) = x² / c² = (c²t₁²) / (c²) = t₁²

Hence we would have the equation of the normal at P given by:

y - (c/t₁) = t₁² ( x - ct₁ )

yt₁ - c = xt₁² - ct₁³

Therefore multiplying throughout by c/t₁² we get:

(ct₁)x -(c/t₁)y = (ct₁)² - (c/t₁)² ...................................... (1)

This can be rewritten as:

x t₁³ - y t₁ - c t₁⁴ + c = 0.

Now if this normal meets the hyperbola again at any other point Q ( c/t₂, c/t₂ ), then:
(ct₂) t₁³ - (c/t₂) t₁ - c t₁⁴ + c = 0

So, t₂ t₁³ - (t₁ / t₂) - t₁⁴ + 1 = 0

∴ ( t₂ t₁³ + 1 ) - (t₁ / t₂)( 1 + t₂ t₁³ ) = 0

∴ ( t₂ t₁³ + 1 ) ( 1 - (t₁ / t₂)) = 0, Here t₁ ≠ t₂

∴ t₂ t₁³ + 1 = 0

∴ t₂ = -1 / t₁³

∴ normal to xy = c² at P( ct₁, c/t₁ ) meets thecurve again at Q ( -c/ t₁³ , -c t₁³ ).
regards
Sumit

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