The common tangent to the circles x2 + y2 = 4 and x2 + y2 + 6x + 8y

Engineering

Mathematics

A Circle

Common Tangent To two Circle

Line and Circle

Question

The common tangent to the circles x² + y² = 4 and x² + y² + 6x + 8y – 24 = 0 also passes through the point

(– 6, 4)

(– 4, 6)

(6, – 2)

(4, – 2)

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Circle x² + y² = 4

⇒ c₁ (0,0) ; r₁ = 2

and circle x² + y² + 6x + 8y – 24 = 0

⇒ c₂ (–3, - 4); r₂ =7

⇒ d = c₁c₂ = 5

also d = |r₁ – r₂|

circles touch internally

Equation of common tangent S₁ – S₂ = 0

⇒ 6x + 8y – 20 = 0

⇒ 3x + 4y – 10 = 0

Point (6, –2) satisfy it.