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Solution


Q.

Let A(–2,–1),B(1,0),C(α,β) and D(γ,δ) be the vertices of a parallelogram ABCD. If the point C lies on 2x – y = 5 and the point D lies on 3x – 2y = 6, then the value of |α+β+γ+δ| is equal to __________.           [2024]

Ans.

(32)

We have, A(–2, –1), B(1, 0), C(α, β) and D(γ, δ) be the vertices of a parallelogram ABCD.

Since, C(α, β) lies on 2x – y = 5

∴   2α - β = 5          ... (i)

Also, D(γ, δ) lies on the line 3x – 2y = 6

So, 3γ - 2δ = 6          ... (ii)

Now, γ + 12 = α - 22  and  δ + 02 = β - 12

γ + 1 = α - 2  and  δ = β - 1

γ = α - 3          ... (iii)

From (ii), 3(α - 3) - 2(β - 1) = 6

Solving (i) and (iii), we get

α = -3; β = -11; γ = -6; δ = -12

∴   |α + β + γ + δ| = |-3 - 11 - 6 - 12| = 32.