Two circles each of the radius 5 units touch each other at the point ( 1,2 ). If the equation of their common tangent is $4 x+3 y=10$, and $C_{1}(\alpha, \beta)$ and $C_{2}(\gamma, \delta), C_{1} \neq C_{2}$ are their centres, then $|(\alpha+\beta)(\gamma+\delta)|$ is equal to ______.

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Circle

Two circles each of the radius 5 units touch each other at the point ( 1,2 ). If the equation of their common tangent is

4 x + 3 y = 10

, and

C_{1} (α, β)

and

C_{2} (γ, δ), C_{1} \neq = C_{2}

are their centres, then

∣ (α + β) (γ + δ) ∣

is equal to ______.

Answer: 40

Solution

The point

(1, 2)

lies on

4 x + 3 y = 10

, so the two centres lie along the unit normal

(\frac{4}{5}, \frac{3}{5})

at distance

5

from it:

C_{1} = (1, 2) + 5 (\frac{4}{5}, \frac{3}{5}) = (5, 5), C_{2} = (1, 2) - 5 (\frac{4}{5}, \frac{3}{5}) = (- 3, - 1) .

Then

α + β = 10

and

γ + δ = - 4

, so

∣ (α + β) (γ + δ) ∣ = ∣10 \cdot (- 4) ∣ = 40.

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Numerical answer

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