Question

Let `AB` be a chord of the circle `x^2+y^2=r^2` subtending a right angle at the center. Then the locus of the centroid of the `Delta PAB` as `P` moves on the circle is (1) A parabola (2) A circle (3) An ellipse (4) A pair of straight lines

Answer 1

`A{rcos(theta+90^0),rsin(theta+90^0)}`
`=A(-rsintheta,rcostheta)`
`theta=0`
`B(r,0),A(0,r)`
Contract of`/_PAB is (alpha,beta)`
`alpha=(r+0+rcostheta)/3=3alpha-r=rcostheta-(1)`
`beta=(0+r+rsintheta)/3=3beta-r=rsintheta-(2)` `(3alpha-r)^2+(3beta-r)^2=r^2cos^2theta+r^2sin^2theta=r^2`
`(alpha-r/3)^2+(beta-r/3)^2=(r/3)^2`
`(r/3,r/3),(r/3)`
option b is correct.