A circle passes through a fixed points O and cuts two perpendicular straight lines OA and OB . If the line AB passes through a fixed point (p , q) , then the locus of the center of the circle OAB is the circle $\frac{p}{x} + \frac{q}{y}$ = ?

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Solution

$x^{2} + y^{2}$ -ax - by = 0 is the circle OAB,
Whose centre is (a/2 , b/2) . The line AB $\frac{x}{a} + \frac{y}{b}$ = 1
passes through (p , q)
$∴ \frac{p}{a} + \frac{q}{b}$ = 1 Put $\frac{a}{2}$ = x, $\frac{b}{2} = y$
$∴ \frac{p}{x} + \frac{q}{y}$ = 2 is the required locus .

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A circle passes through a fixed points O and cuts two perpendicular straight lines OA and OB . If the line AB passes through a fixed point (p , q) , then the locus of the center of the circle OAB is the circle px+qy = ?

x2+y2 -ax - by = 0 is the circle OAB,Whose centre is (a/2 , b/2) . The line AB xa+yb = 1passes through (p , q)∴pa+qb = 1 Put a2 = x, b2=y∴px+qy = 2 is the required locus .

A circle passes through a fixed points O and cuts two perpendicular straight lines OA and OB . If the line AB passes through a fixed point (p , q) , then the locus of the center of the circle OAB is the circle $\frac{p}{x} + \frac{q}{y}$ = ?

$x^{2} + y^{2}$ -ax - by = 0 is the circle OAB,
Whose centre is (a/2 , b/2) . The line AB $\frac{x}{a} + \frac{y}{b}$ = 1
passes through (p , q)
$∴ \frac{p}{a} + \frac{q}{b}$ = 1 Put $\frac{a}{2}$ = x, $\frac{b}{2} = y$
$∴ \frac{p}{x} + \frac{q}{y}$ = 2 is the required locus .