The equation of the locus of the mid-points of chords of the circle `4x^2 + 4y^2-12x + 4y +1= 0` that subtends an angle of at its centre is `(2pi)/3` at its centre is `x^2 + y^2-kx + y +31/16=0` then k is

Explanation

Given, `4x^(2)+4y^(2)-12x+4y+1=0`
`rArrx^(2)+y^(2)-3x+y+1//4=0`
Whose centre is `((3)/(2),(1)/(2))` and radius
`=sqrt(((3)/(2))^(2)+(-(1)/(2))^(2)-(1)/(4))=sqrt((9)/(4)+(1)/(4)-(1)/(4))`
`=(3)/(2)`
Again, let S be a circle with centre at C and AB is given chord and AD substendf angle `2pi//3` at the centre and D be the mid-point of AB and let its coordinates are (h,k).
Now, " " `angleDCA=(1)/(2)(angleBCA)=(1)/(2)*(2pi)/(3)=(pi)/(3)`
Using sine rule in `DeltaADC`,
`(DA)/(sinpi//3)=(CA)/(sinpi//2)`
`rArr DA=CAsinpi//3=(3)/(2)*(sqrt3)/(2)`
Now, in `DeltaACD`
`CD^(2)=CA^(2)-AD^(2) = (9)/(4)-(27)/(16)=(9)/(16)`
But" "`CD^(2)=(h-3//2)^(2)+(k+1//2)^(2)`
`rArr (h-3//2)^(2)+(k+1//2)^(2)=(9)/(16)`
Hence, locus of a point is
`(x-(3)/(2))^(2)+(y+(1)/(2))^(2)=(9)/(16)`
`rArr 16x^(2)+ 16y^(2)-48x+16y+31=0`