Question

Find the angle between the circles are x2 + y2 + 6x – 10y – 135 = 0, x2 + y2 – 4x + 14y – 116 = 0.

Answer 1

ANSWER :-

C1 = (–3, 5)

C2 = (2, –7) 

r1 = √(9 + 25 + 135)  r2 = √(4 + 49 + 116) 

r1 = 13  r2 = 13

C1C2 = √((−3 − 2)2 + (5 + 7)2) = 13

Answer 2

The angle between the given circles is 60°

Given,

Circle 1 ⇒ x² + y² + 6x – 10y – 135 = 0

Circle 2 ⇒ x² + y² – 4x + 14y – 116 = 0

To Find,

Angle between the circles

Solution,

General circle equation -

x² + y² + 2gx +2fy + c = 0

Comparing we get -

g₁ = 3, f₁ = -5, and c₁ = -135

g₂ = -2, f₂ = 7, and c₂ = -116

Center of Circle 1 = C₁ = (-g₁, -f₁) = (-3, 5)

Center of Circle 2 = C₂ = (-g₂, -f₂) = (2, -7)

Radius = $\sqrt{g^2+f^2-c}$

r₁ = $\sqrt{3^2+(-5)^2+135} = \sqrt{169}$ = 13 units

r₂ = $\sqrt{(-2)^2+7^2+116} = \sqrt{169}$ = 13 units

Distance between the centers of the circle = C₁C₂

C₁C₂ = d = $\sqrt{(-3-2)^2+(5+7)^2}$

d = $\sqrt{169}$ = 13 units

Now angle between the circles is given by -

Cos θ = [(r₁)² + (r₂)² - (d)²]/(2 * r₁ * r₂)

Cos θ = $\frac{13^2+13^2-13^2}{2*13*13}$

Cos θ = $\frac{1}{2}$

θ = Cos ⁻¹ ( $\frac{1}{2}$ )

θ = 60°

Therefore, the angle between the given circles is 60°

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