The common tangents to the circles `x^2 + y^2 + 2x =0 and x^2 + y^2-6x=0` form a triangle which is

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

B

Explanation

Centres of two circles are `C_(1)(-1, 0) and C_(2)(3, 0)` respectively .
Clearly, `C_(1)C_(2)=` Sum of the radii.
So, two circles touch each other externally.

Let P be the point of intersection of transverse common tangents. Then, P divides `C_(1)C_(2)` externally in the ratio 1:3 . So, coordinates of P are (-3, 0).
The equation of any line passing through P (-3, 0) is
y=m(x+3)
If it touches the circle `x^(2)+y^(2)+2x=0`, then
`|(2m)/(sqrt(m^(2)+1))|=1 rArr m = pm (1)/(sqrt(3))`
`:. APB = 60^(@)`
Since `Delta PAB` is isosceles. Therefore , `angleA=angle B=60^(@)`
Hence, `DeltaPAB` is equilateral.