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

The Correct Option is C

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:

\[ = 155 - 118 \]

\[ = 37 \]