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:
- 5
- -27
- 37
- 437
The Correct Option is C
Solution and Explanation
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:
\[ = 155 - 118 \]
\[ = 37 \]
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