1. ## Circle question

A moving point P is such that the length of the tangent from P to the circle x^2+y^2=16 is equal to the distance of P from point (8,8). Show that the locus of P is the straight line x+y=9.

Can anyone help me? Thanks...

2. Originally Posted by MichaelLight
A moving point P is such that the length of the tangent from P to the circle x^2+y^2=16 is equal to the distance of P from point (8,8). Show that the locus of P is the straight line x+y=9.

Can anyone help me? Thanks...
1. Draw a sketch!

2. Let P(x, y) denote a point whose distance to Q(8, 8) equals the tangent segment to the given circle. The you know that the distance d can be calculated by:

$d^2 = (x-8)^2+(y-8)^2$

3. The distance d, the radius r of the circle and the distance of P to the center of the circle form a right triangle:

$d^2+r^2=x^2+y^2$

4. Plug in r = 4 and d² from the 1st equation into the 2nd equation:

$(x-8)^2+(y-8)^2+16=x^2+y^2$

Solve for y.

3. Originally Posted by MichaelLight
A moving point P is such that the length of the tangent from P to the circle x^2+y^2=16 is equal to the distance of P from point (8,8). Show that the locus of P is the straight line x+y=9.

Can anyone help me? Thanks...
draw a circle C of radius 0 at the point (8,8). Then the distance of P from (8,8) is the length of the tangent drawn from P to C.
So the locus of P is the radical axis of the circles ${(x-8)}^2+{(y-8)}^2=0$ and $x^2+y^2=16$.
we know that radical axis of any two cirlces is a straight line.