A chord of the circle `x^(2)+y^(2)=a^(2)` cuts it at two points A and B such that `angle AOB = pi //2`, where O is the centre of the circle. If there is a moving point P on this circle, then the locus of the orthocentre of `DeltaPAB` will be a

Options

A. parabola
B. circle
C. straight line
D. none of these

Answer

B

Explanation

Let the coordinates of A and B be (a, 0) and (0, a) respectively. Further. Let P (h,k) be a point on the circle.

Clearly, circumcentre of `DeltaPAB` is at the origin and the centroid is `((h+a)/(3), (k+a)/(3))`.
Let `(alpha, beta)` be the orthocentre. Then,
`(h+a)/(3)=(2xx0+1xxalpha)/(2+1)` and , `(k+a)/(3)=(2xx0+1xxbeta)/(3)`
`rArr alpha = h + a and beta = k + a`
`rArr (h=alpha-a and k=beta-a`
Since(h, k) lies on `x^(2)+y^(2)=a^(2)`.
`:. h^(2)+k^(2)=a^(2)rArr (alpha-a)^(2)+(beta-a)^(2)=a^(2)`
Hence, the locus of `(alpha , beta)` is `(x-a)^(2)+(y-a)^(2)=a^(2)`, which represents a circle.