Match the following lists.

Answer

`a rarr r; b rarr s; c rarr q; d rarr q`

Explanation

a. Since `(2,3)` lies inside the circle, such chord is bisected at (2,3) , which has equation
`y-3 = -x(x-2)`
or `x+y-5=0` or `a=b=1`
b. Let P be the point `(alpha, beta)`. Then `alpha^(2)+beta^(2)+2alpha +2beta =0`
The midpoint of OP is `( alpha //2, beta//2)` .
Therefore, the locus of `(alpha //2 , beta//2)` is
`4x^(2)+4y^(2)+4x+4y=0`
i.e., `2g=1,2f=1`
and `g+f=1`
c. The centes of the circles are (1,2) and (5,-6)
The equation of `C_(1)C_(2)` is
`y-2= - (8)/(4) (x-1)`
i.e., `2x+y-4=0`
The equation of the radical axis is
`8x-16y-56=0`
i.e., `x-2y-7=0`
The point of intersection is `(3,-2)`.
d. If `theta` is the angle between the tangents, then
`(theta)/(2) =("Radius")/("Distance between "(3sqrt(3),3)"and"(0,0))=(1)/(2)`
or `(theta)/(2)=(pi)/(6)` or `theta=(pi)/(3)`or `2sqrt(3) tan theta=6`