The circle `C_(1):x^(2)+y^(2)=3`, with centre at O, intersects the parabola `x^(2)=2y` at the point P in the quadrant. Let the tangent to the circle `C_(1)` at P touches other tqo circles `C_(2)andC_(3)` at `R_(2)andR_(3)`, respectively. Suppose `C_(2)andC_(3)` have equal radii `2sqrt(3)` and centres `Q_(2)andQ_(3)`. respectively. If `Q_(2)andQ_(3)` lie on the y-axis, then

Options

A. `Q_(2)Q_(3)= 12`
B. `R_(2)R_(3)= 4sqrt6`
C. area of the `DeltaPR_(2)R_(3) " is " 6sqrt2`
D. area of the `DeltaPQ_(2)Q_(3)" is " 4sqrt2`

Answer

A, B, C

Explanation

Given `C_(1):x^(2)+y^(2)=3` intersects the parabola `x^(2)=2y`.

On solving `x^(2)+y^(2)=3 and x^(2)=2y`, we get
`y^(2)+2y=3`
`rArr y^(2)+2y-3=0`
`rArr (y+3)(y-1)=0`
`therefore y=1, -3["neglecting " y=-3, as -sqrt3leylesqrt3]`
`therefore y=1 rArrx=pmsqrt2`
`rArr P(sqrt2,1) in "I quadrant"`
Equation of tangent at `Psqrt2,1)"to " C_(1):x^(2)+y^(2)=3` is
`sqrt2x+1*y=3" "...(i)`
Now, let the centres of `C_(2) and C_(3)` at `R_(2) and R_(3)` shown as below

Let `Q_(2)` be (0,k) and radius is `2sqrt3`.
`therefore (|0+k-3|)/(sqrt(2+1))=2sqrt3`
`rArr |k-3|=6`
`rArr h=9, -3`
`therefore Q_(2)(0,9) and Q_(3)(0,-3)`
Hence, `Q_(2)Q_(3)=12`
`therefore` option (a) is correct.
Also, `R_(2)R_(3)` is common internal tangent to `C_(2) and C_(3)`,
and `r_(2)=r_(3)=2sqrt3`
`R_(2)R_(3)=sqrt(d^(2)-(r_(1)+r_(2
))^(2))=sqrt(12^(2)-(4sqrt3)^(2))`
`=sqrt(144-48)=sqrt96=4sqrt6`

`therefore` (b) is correct.
`therefore` Length of perpendicular from `O(0,0) "to " R_(2)R_(3)` is equal to radius of `C_(1)=sqrt3`
`therefore " Area of " DeltaPQ_(2)Q_(3)=(1)/(2)Q_(2)Q_(3)xxsqrt2=(sqrt2)/(2)xx12=6sqrt2`
`therefore` Option (c) is correct.
Also, area of `DeltaPQ_(2)Q_(3)=(1)/(2)Q_(2)Q_(3)xxsqrt2=(sqrt2)/(2)xx12=6sqrt2`
`therefore` Option (d) is incorrect.