If the locus of the point, whose distance from the point (2, 1) and (1, 3) are in the ratio 5 : 4, is , then the value of is equal to : [2024]

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Solution

Q.
If the locus of the point, whose distance from the point (2, 1) and (1, 3) are in the ratio 5 : 4, is $a x^{2} + b y^{2} + c x y + d x + e y + 170 = 0$ , then the value of $a^{2} + 2 b + 3 c + 4 d + e$ is equal to : [2024]

1 5

2 – 27

3 37

4 437

Ans.
(3)

Let given point be (h, k) whose distance from (2, 1) and (1, 3) is in ratio 5 : 4.

$\Rightarrow \frac{{(h - 2)}^{2} + {(k - 1)}^{2}}{{(h - 1)}^{2} + {(k - 3)}^{2}} = \frac{25}{16}$

$\Rightarrow \frac{h^{2} + k^{2} - 4 h - 2 k + 5}{h^{2} + k^{2} - 2 h - 6 k + 10} = \frac{25}{16}$

On replacing h by x and k by y, we get

$16 x^{2} + 16 y^{2} - 64 x - 32 y + 80 = 25 x^{2} + 25 y^{2} - 50 x - 150 y + 250$

$\Rightarrow 9 x^{2} + 9 y^{2} + 14 x - 118 y + 170 = 0$

On comparing, we get

$a = 9, b = 9, c = 0, d = 14, e = - 118$

$∴ a^{2} + 2 b + 3 c + 4 d + e = 81 + 18 + 56 - 118 = 37$

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