Question

If the locus of the point, whose distances from the point $(2, 1)$ and $(1, 3)$ are in the ratio $5 : 4$, is $ax^2 + by^2 + cxy + dx + ey + 170 = 0,$ then the value of $a^2 + 2b + 3c + 4d + e$ is equal to:

• A. 5
• B. -27
• C. 37
• D. 437

Answer 1

Answer: (C)

37

Answer 2

Let $P(x, y)$

$\frac{(x - 2)^2 + (y - 1)^2}{(x - 1)^2 + (y - 3)^2} = \frac{25}{16}$

Expanding and simplifying:

$9x^2 + 9y^2 + 14x - 118y + 170 = 0$

From the equation:

$a^2 + 2b + 3c + 4d + e = 81 + 18 + 0 + 56 - 118$

Calculating:

$= 155 - 118$

$= 37$