Find locus of mid point of chord of x2a2 - y2b2 - 1 which subtend 90o at the centre -

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Standard XI

Mathematics

Terminologies Related to a Hyperbola

Find locus of...

Question

Find locus of mid point of chord of $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ which subtend $90^{o}$ at the centre ?

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Solution

$t_{1} t_{2} = - 1$ {because PQ as focal cord}
$x_{2} = 2 a + a (t_{1}^{2} + t_{2}^{2} + t_{1} t_{2})$
$y_{2} = - a t_{1} t_{2} (t_{1} + t_{2})$
$x_{2} = 2 a + a (t_{1}^{2} + t_{2}^{2} - 1); y_{2} = - a t_{1} t_{2} (t_{1} + t_{2})$
$y_{2} = a (t_{1} + t_{r})$
So, $y_{1} = y_{2}$

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Find locus of mid point of chord of x2a2−y2b2=1 which subtend 90o at the centre ?

t1t2=−1 {because PQ as focal cord}x2=2a+a(t21+t22+t1t2)y2=−at1t2(t1+t2)x2=2a+a(t21+t22−1);y2=−at1t2(t1+t2)y2=a(t1+tr)So, y1=y2

Find locus of mid point of chord of $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ which subtend $90^{o}$ at the centre ?

$t_{1} t_{2} = - 1$ {because PQ as focal cord}
$x_{2} = 2 a + a (t_{1}^{2} + t_{2}^{2} + t_{1} t_{2})$
$y_{2} = - a t_{1} t_{2} (t_{1} + t_{2})$
$x_{2} = 2 a + a (t_{1}^{2} + t_{2}^{2} - 1); y_{2} = - a t_{1} t_{2} (t_{1} + t_{2})$
$y_{2} = a (t_{1} + t_{r})$
So, $y_{1} = y_{2}$