Question

If PQR is the triangle formed by the common tangent to the circle x² + y² + 6x = 0 and x² + y² – 2x = 0 then –

• A. Centroid of DPQR is (1, 0)
• B. Incentre of the DPQR is (1, 1)
• C. Circum centre of the DPQR is (1, 2)
• D. Orthocentre of the DPQR is (2, 0)

Answer 1

Answer: (A)

Centroid of DPQR is (1, 0)

Answer 2

C₁ = (– 3, 0), r₁ = 3 , C₂ = (1, 0), r₂ = 1

C₁C₂ = r₁ + r₂ So both circles touches each other externally at the origin. Common tangent at the origin is y-axis

y = mx + c be a direct common tangent to two circles = ฑ 3 and = ฑ 1

– 6m c + c² = 9  2 cm + c² = 1

cm = – 1 and c² = 3  c = ฑ $\sqrt { 3 }$

Equation of common tangent are

y = $\sqrt { 3 }$, y = $\sqrt { 3 }$, x = 0

Both the lines makes an angle of 60⁰ with x = 0 So the triangle formed by these lines is equilateral so that centroid, circumcentre, orthocentre are coincide with its incentre (1, 0), centre of smaller circle inscribed in the triangle PQR.