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If a circle passes through the point (a,...

If a circle passes through the point `(a, b)` and cuts the circle `x^2 +y^2=k^2` orthogonally, then the equation of the locus of its center is

A

`2ax+2by- (a^(2)+b^(2)+k^(2))=0`

B

`2ax+2by- (a^(2)-b^(2)+k^(2))=0`

C

`x^(2)+ y^(2)-3ax - 4by + a^(2) + b^(2) -k^(2) = 0 `

D

`x^(2)+ y^(2)-2ax - 3by + (a^(2) - b^(2) -k^(2)) = 0 `

Text Solution

Verified by Experts

The correct Answer is:
A
Let `x^(2)+y^(2)+2gx+2fy+c=0," cuts "x^(2)+y^(2)=k^(2)` orthogonally.
`rArr 2g_(1)g_(2)+2f_(1)f_(2)=c_(1)+c_(2)`
`rArr -2g*0-2f*0=c-k^(2)`
`rArr c=k^(2)" " ...(i)`
Also, `x^(2)+y^(2)+2gx+2fy+k^(2)=0` passes through (a, b).
`x^(2)+y^(2)+2gx+2fy+k^(2)=0" "...(ii)`
`rArr` Required equation of locus of centre is
`-2ax-2by+a^(2)+b^(2)+k^(2)=0`
or `2ax+2by-(a^(2)+b^(2)+k^(2))=0`
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