Question

Find the radical centre of circles x² +y² +3x+2y+1 = 0,x² +y² –x+6y+5 = 0 and x² +y² +5x–8y+15= 0.

Also find the equation of the circle cutting them orthogonally.

Answer 1

Let S₁ ≡ x² +y² +3x+2y+1 = 0

S₂ ≡ x² +y² –x+6y+5 = 0

S₃ ≡ x² +y² +5x–8y+15 = 0

Equations of radical axis

S₁ –S₂ = 0 ⇒ 4x–4y–4 = 0 or x–y–1 = 0___________(1)

S₂ –S₃ = 0 ⇒ –6x+14y–10 = 0 or –3x+7y–5 = 0 ________(2)

Solve equations (1) & (2) we get (3, 2) as radical centre.

(1) ×3–(2)×1

– 3x +7y – 5= 0

– 3x+ 3y+ 3 = 0

y = 2

x–2–1= 0 ⇒ x = 3

radius of fourth circle cutting these three circles orthogonally is length of tangent from this centre to any one circle

∴r = \(\sqrt{s_1}\)

\(=\sqrt{3^2+2^2+3.3+2.2+1}\)

\(=\sqrt{9+4+9+4+1}\)

\(=\sqrt{27}\)

\(=3\sqrt{3}\)

∴ Equation of circle is (x–3)² +(y–2)² = \((3\sqrt{3})^2\)

x² +y² –6x–4y–14 = 0