Angle between the common tangents of the hyperbolas x 2- a 2- y 2- b 2-1 and y 2- a 2- x 2- b 2-1- A- 1-2ftB- -C-D-

The correct option is A π2Equation of the tangent x^2a^2 y^2b^2=1 is y=mx±√(a^2m^2 b^2) Equation of the tangent y^2a^2 x^2b^2=1 or x^2 b^2 y^2 a^2=1 is nbs ...

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Standard XII

Mathematics

Chord of Contact: Hyperbola

Angle between...

Question

Angle between the common tangents of the hyperbolas $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$

\frac{π}{2}

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\frac{π}{3}

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\frac{π}{4}

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\frac{π}{6}

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Solution

The correct option is A $\frac{π}{2}$
Equation of the tangent $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is $y = m x \pm \sqrt{a^{2} m^{2} - b^{2}}$

Equation of the tangent $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1 or \frac{x^{2}}{- b^{2}} - \frac{y^{2}}{- a^{2}} = 1$ is
$y = m x \pm \sqrt{(- b^{2}) m^{2} - (- a^{2})}$

Since tangents are common,
$∴ - b^{2} m^{2} + a^{2} = a^{2} m^{2} - b^{2} \Rightarrow m^{2} = 1 \Rightarrow m = \pm 1$
$∴$ Angle between common tangents is $\frac{π}{2}$

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Angle between the common tangents of the hyperbolas x2a2−y2b2=1 and y2a2−x2b2=1

The correct option is A π2Equation of the tangent x2a2−y2b2=1 is y=mx±√a2m2−b2 Equation of the tangent y2a2−x2b2=1 or x2−b2−y2−a2=1 is y=mx±√(−b2)m2−(−a2) Since tangents are common, ∴−b2m2+a2=a2m2−b2⇒m2=1⇒m=±1 ∴ Angle between common tangents is π2

Angle between the common tangents of the hyperbolas $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$