A rod of length eight units moves such that its ends A and B always lie on the lines x – y + 2 = 0 and y + 2 = 0, respectively. If the locus of the point P, that divides the rod AB internally in the ration 2 : 1 is , then is equal to: [2025]

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Solution

Q.
A rod of length eight units moves such that its ends A and B always lie on the lines x – y + 2 = 0 and y + 2 = 0, respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2 : 1 is $9 (x^{2} + α y^{2} + β x y + γ x + 28 y) - 76 = 0$ , then $α - β - γ$ is equal to: [2025]

1 22

2 21

3 24

4 23

Ans.
(4)

Let P(h, k) be the point which divides AB internally in the ratio 2 : 1.

$∴ h = \frac{2 β + α}{3} and k = \frac{- 4 + α + 2}{3}$

$\Rightarrow α = 3 k + 2$

$∴ 2 β = 3 h - α = 3 h - 3 k - 2$

So, AB = 8

$\Rightarrow {(α - β)}^{2} + {(α + 4)}^{2} = 64$

$\Rightarrow (3 k + 2 - {(\frac{3 h - 3 k - 2}{2}))}^{2} + {(3 k + 2 + 4)}^{2} = 64$

$\Rightarrow \frac{{(9 k - 3 h + 6)}^{2}}{4} + {(3 k + 6)}^{2} = 64$

$\Rightarrow 9 [{(3 k - h + 2)}^{2} + 4 {(k + 2)}^{2}] = 64 \times 4$

$\Rightarrow 9 (9 k^{2} + h^{2} + 4 - 6 k h + 12 k - 4 h + 4 k^{2} + 16 + 16 k) = 256$

$\Rightarrow 9 (13 k^{2} + h^{2} - 6 k h + 28 k - 4 h) = 76$

$\Rightarrow 9 (x^{2} + 13 y^{2} - 6 x y - 4 x + 28 y) = 76$

Comparing the equation with

$[9 (x^{2} + α y^{2} + β x y + γ x + 28 y) - 76] = 0$

$\Rightarrow α = 13, β = - 6, γ = - 4$

$∴ α - β - γ = 13 + 6 + 4 = 23$

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