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21/01/2022 2:24 pm
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Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and C1 (α, β) and C2 (γ, δ), C1 ≠ C2 are their centres, then |(α + β) (γ + δ)| is equal to ___________.
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21/01/2022 2:32 pm
Slope of line joining centres of circles = \(\frac{4}{3}\) = tan θ
⇒ cos θ = \(\frac{3}{5}\), sin θ = \(\frac{4}{5}\)
Now using parametric form
\(\frac{x - 1}{cos θ}\) = \(\frac{y - 2}{sin θ}\) = ±5
(x, y) = (1 + 5cos θ, 2 + 5sinθ)
(α, β) = (4, 6)
(x, y) = (γ, δ) = (1 - 5cos θ, 2 - 5 sinθ)
((γ, δ) = (-2, -2)
⇒ |(α + β) (γ + δ)| = |10x - 4| = 40