Tangent to circle

**arze** · May 30th 2010, 11:18 PM

The line pair joining the origin to the points A and B of intersection of the conic $ax^2+by^2=1$ and $lx+my=1$ is
$(a-l^2)x^2-2lmxy+(b-m^2)y^2=0$
If angle AOB is a right angle, show that AB touches the circle
$(a+b)(x^2+y^2)=1$
I have no idea how to do this. Any pointers?
Thanks a million!

**sa-ri-ga-ma** · May 31st 2010, 02:31 AM

If angle AOB is a right angle,

$(a-l^2) + (b-m^2) = 0$

Or $a+b = l^2 + m^2$

If lx + my = 1 is the tangent to a circle, the distance of this line from the center is equal to the radius.

$\frac{1}{l^2 + m^2} = r^2$

$\frac{1}{a+b} = r^2$

So the equation of the circle is

$x^2 + y^2 = \frac{1}{a+b}$

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