Question

​​A chord PQ of the hyperbola xy = c2 is tangent to the hyperbola x2/a2 - y2/b2 = -1. Find the locus of the middle point of PQ.

Answer 1

xy=c2 is a transformed form of x2−y2=a2where c=a2√⇒a=c2‾√⇒a2=2c2Hence xy=c2 is written asx2−y2=2c2Let mid point be (h,k), hence equation of chord is T=S1xh−yk=h2−k2⇒yk=xh+(k2−h2)⇒y=xhk+(k2−h2)kCompare with y=mx+cm=hk and c=k2−h2kNow y=mx is tangent to x2a2−y2b2=−1 isc2=b2−a2m2Hence locus is(k2−h2k)2=b2−a2 h2k2⇒(k2−h2)2=k2b2−a2h2h→x and k→y⇒(y2−x2)2=y2b2−a2x2

Answer 2

The locus of the middle point of PQ is $(y^2-x^2)^2=y^2b^2-a^2x^2$

Step-by-step explanation:

Given:

Here it is given that a chord PQ of the hyperbola $xy=c^2$ is tangent to the hyperbola $\frac{x^2}{a^2}- \frac{y^2}{b^2}=-1$.

$xy=c^2$ is a transformed form of $x^2-y^2=a^2$

where $c=\frac{a}{\sqrt{2} }$

⇒ $a=c\sqrt{2}$

⇒ $a^2=2c^2$

Hence $xy=c^2$ is written as

$x^2-y^2=2c^2$

Let mid point be (h,k), hence equation of chord is,

$T=S_1$

⇒ $xh-yk=h^2-k^2$

⇒ $xh-h^2+k^2=yk$

⇒ $yk=xh+(k^2-h^2)$

⇒ $y=\frac{xh}{k}+\frac{k^2-h^2}{k}$

Compare with y=mx+c, we get

$m=\frac{h}{k} and c=\frac{k^2-h^2}{k}$

Now y=mx is tangent to $\frac{x^2}{a^2}- \frac{y^2}{b^2}=-1$ is,

$c^2=b^2-a^2m^2$

Hence locus is,

$(\frac{k^2-h^2}{k})^2 =b^2-a^2\frac{h^2}{k^2}$

⇒ $(k^2-h^2)^2=k^2b^2-\frac{a^2h^2}{k^2}$

⇒ $(k^2-h^2)^2=k^2b^2-a^2h^2$

h → x and k → y

$(y^2-x^2)^2=y^2b^2-a^2x^2$

Therefore, the locus of the middle point of PQ is $(y^2-x^2)^2=y^2b^2-a^2x^2$