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:
$\displaystyle d^2 = (x-8)^2+(y-8)^2$
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:
$\displaystyle d^2+r^2=x^2+y^2$
4. Plug in r = 4 and dē from the 1st equation into the 2nd equation:
$\displaystyle (x-8)^2+(y-8)^2+16=x^2+y^2$
Solve for y.
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 $\displaystyle {(x-8)}^2+{(y-8)}^2=0$ and $\displaystyle x^2+y^2=16$.
we know that radical axis of any two cirlces is a straight line.