Consider two circles `S, =x^2+y^2 +8x=0 and S_2=x^2+y^2-2x=0`. Let `DeltaPOR` be formed by the common tangents to circles `S_1 and S_2`, Then which of the following hold(s) good?

Options

A. Incentre of `DeltaPQR` is (1,0)
B. The equation of radical axis of circles `S_(1)` and `S_(2)` is `y = 0`
C. Product of slope of direct common tangents is `(16)/(9)`
D. If transverse common tangent intersects direct common tangents at points A and B, then AB equals 4.

Answer

A, D

Explanation

We have `S_(1) -= x^(2)+y^(2) +8x = 0` with centre `(-4,0)` and radius `=4`.
Also, we have `S_(2) -= x^(2)+y^(2) -2x =0` with centre (1,0) and radius =1,
Hence, circles `S_(1)` and `S_(2)` touch each other externally. Also, `P ((8)/(3),0)`
(As P divides the line segment joining the centres in a ratio of the radii externally)
Let equation of the direct common tangent by `y = m (x-(8)/(3))` By applying condition of tangency, we get
`(|m-(8m)/(3)|)/(sqrt(1+m^(2))) =1`
`rArr m^(2) = (9)/(16)` or `m = +- (3)/(4)`