coordinate geometry

**prasum** · July 20th 2010, 02:29 AM

find the locus of centres of circle x^2+y^2-2ax-2by+2=0 where a and b are parameters if the tangents from any to each of the circles are orthogonal

**Soroban** · July 20th 2010, 02:00 PM

Hello, prasum!

I'm not sure I understand the problem . . .

Find the locus of centres of circle: $x^2+y^2-2ax-2by+2\:=\:0$
. . where $a$ and $b$ are parameters,
if the tangents from any point to each of the circles are orthogonal.

The circle has the equation: . $(x-a)^2 + (y-b)^2 \:=\:a^2+b^2-2$

It has center $C(a,b)$ and radius $r\,=\,\sqrt{a^2+b^2-2}$

Let $P(h,k)$ be any point (exterior to the circle).

Tangents are drawn from $P$ to $A$ and $B$ on the circle.
. . The tangents are orthogonal: . $\angle P = 90^o$

Code:

    |
    |         * * *
    |     *           * A
    |   *               ♥
    |  *              o  *o
    |               o       o
    | *       C   o       *   o    P
    | *    (a,b)♥         *     ♥(h,k)
    | *           o       *   o
    |               o       o
    |  *              o  *o
    |   *               ♥
    |     *           * B
    |         * * *
- - + - - - - - - - - - - - - - - - - - - - - -
    |

Since $\angle A = \angle B = \angle P = 90^o,\;PA = PB,\:CA = CB$
. . then quadrilateral $APBC$ is a square.

Its diagonal is: . $CP \,=\,\sqrt{2}\,r \;=\;\sqrt{2(a^2+b^2-2)}$

Therefore, the locus of the centers of the circles
. . is a circle with center $(h,k)$ and radius $\sqrt{2(a^2+b^2-2)}$

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