If x²/a² + y²/b² = 1 (a > b) and x² - y² = c² cut at right angles, then

Correct Answer is : a2−b2a² - b²a2−b2 = 2c22c²2c2

Correct Answer is : a2−b2a² - b²a2−b2 = 2c22c²2c2

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VITEEE 2012 Paper

Section: Mathematics

Question : 108 of 120

Marks: +1, -0

\frac{x^2}{a^2} + \frac{y^2}{b^2}

x^2 - y^2

c^2

cut at right angles, then

Solution:

\frac{x^2}{a^2} + \frac{y^2}{b^2}

= 1 ...(i)

On differentiating w.r.t. x, we get

\frac{2x}{a^2} + \frac{2y}{b^2} \cdot \frac{dy}{dx}

= 0

⇒

\frac{dy}{dx}

= -

\frac{x b^2}{a^2 y}

and

x^2 - y^2

c^2

On differentiating w.r.t. x, we get

2x - 2y

\frac{dy}{dx}

= 0

⇒

\frac{dy}{dx}

\frac{x}{y}

The two curves will cut at right angles, if

\left(\frac{dy}{dx}\right)_{c_1}

\left(\frac{dy}{dx}\right)_{c_2}

= - 1

⇒ -

\frac{b^2 x}{a^2 y}

\frac{x}{y}

= - 1

⇒

\frac{x^2}{a^2}

\frac{y^2}{b^2}

⇒

\frac{x^2}{a^2}

\frac{y^2}{b^2}

\frac{1}{2}

[using eq. (i)]

On substituting these values in

x^2 - y^2

x^2

, we get

\frac{a^2}{2} - \frac{b^2}{2}

c^2

a^2 - b^2

2c^2

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