Equation of Tangent to Hyperbola in all Forms

Equation of Tangent to Hyperbola \(x^2\over a^2\) – \(y^2\over b^2\) = 1

(a) Point form :

The equation of tangent to the given hyperbola at its point (\(x_1, y_1\)) is

\(x{x_1}\over a^2\) – \(y{y_1}\over b^2\) = 1

Example : Find the equation of tangent to the hyperbola \(16x^2\) – \(9y^2\) = 144 at (5, 16/3).

Solution : We have, \(16x^2\) – \(9y^2\) = 144

\(\implies\) \(x^2\over 9\) – \(y^2\over 16\) = 1

Compare given equation with \(x^2\over a^2\) – \(y^2\over b^2\) = 1

a = 3 and b = 16

Hence, required equation of tangent is \(5x\over 9\) – \(16/3\over 16\) = 1

= \(5x\over 9\) – \(y\over 3\) = 1

(b) Slope form :

The equation of tangent to the given hyperbola whose slope is ‘m’, is

y = mx \(\pm\) \(\sqrt{a^2m^2 – b^2}\)

The Point of contact are (\({\mp} a^2m\over \sqrt{a^2m^2 – b^2}\), \({\mp} b^2\over \sqrt{a^2m^2 – b^2}\))

Note that there are two parallel tangents having the same slope m.

Example : Find the tangent to the hyperbola \(x^2\over 25\) – \(y^2\over 16\) = 1 whose slope is 1.

Solution : We have, \(x^2\over 25\) – \(y^2\over 16\) = 1

Compare given equation with \(x^2\over a^2\) – \(y^2\over b^2\) = 1

a = 5 and b = 4

Hence, required equation of normal is y = x \(\pm\) \(\sqrt{9}\)

\(\implies\) y = x \(\pm\) 3

(c) Parametric form :

The equation of tangent to the given hyperbola at the point (asec\(\theta\), btan\(\theta\)), is

\(xsec\theta\over a\) – \(ytan\theta\over b\) = 1

Example 3 : Find the tangent to the hyperbola \(x^2 – 4y^2\) = 36 which is perpendicular to the line x – y + 4 = 0

Solution : Let m be the slope of the tangent, since the tangent is perpendicular to the line x – y = 0

\(\therefore\)   m\(\times\)1 = -1 \(\implies\) m = -1

Since \(x^2-4y^2\) = 36 or \(x^2\over 36\) – \(y^2\over 9\) = 1

Comparing this with \(x^2\over a^2\) – \(y^2\over b^2\) = 1

\(\therefore\)   \(a^2\) = 36 and \(b^2\) = 9

So the equation of the tangent are y = -1x \(\pm\) \(\sqrt{36\times {-1}^2 – 9}\)

\(\implies\) y = x \(\pm\) \(\sqrt{27}\) \(\implies\) x + y \(\pm\) 3\(\sqrt{3}\) = 0

Hope you learnt equation of tangent to hyperbola in all forms, learn more concepts of hyperbola and practice more questions to get ahead in the competition. Good luck!

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