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Question: For a given non-zero value of m each of the lines \(\frac{x}{a} - \frac{y}{b}\)= m and \(\frac{x}{a}...

For a given non-zero value of m each of the lines xa−yb\frac{x}{a} - \frac{y}{b}= m and xa+yb\frac{x}{a} + \frac{y}{b}= m meets the hyperbola x2a2−y2b2\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}=1 at a point. Sum of the ordinates of these points, is

A

a(1+m2)m\frac{a(1 + m^{2})}{m}

B

b(1−m2)m\frac{b(1 - m^{2})}{m}

C

0

D

a+b2m\frac{a + b}{2m}

Answer

0

Explanation

Solution

Ordinate of the point of intersection of the line xa−yb\frac{x}{a} - \frac{y}{b}= m and the hyperbola is given by (xa−yb)(xa−yb+2yb)\left( \frac{x}{a} - \frac{y}{b} \right)\left( \frac{x}{a} - \frac{y}{b} + \frac{2y}{b} \right) = 1 i.e. m (m+2yb)\left( m + \frac{2y}{b} \right)= 1 i.e.

y = b(1−m2)2m\frac{b(1 - m^{2})}{2m}

Similarly ordinate of the point of intersection of the line xa+yb\frac{x}{a} + \frac{y}{b}= m and the hyperbola is given by y = b(m2−1)2m\frac{b(m^{2} - 1)}{2m}

\ Sum of the ordinates is 0.