Key Concepts and Formulas

• Locus: The set of all points satisfying a given condition.
• Distance Formula: The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• Apollonius Circle: The locus of a point that moves such that the ratio of its distances from two fixed points is constant.

Step-by-Step Solution

Step 1: Define the General Point and Fixed Points

Let $P(x, y)$ be a general point on the locus. We are given two fixed points $A(2, 1)$ and $B(1, 3)$. We need to find the equation satisfied by $(x, y)$.

Step 2: Express the Given Condition Using the Distance Formula

The problem states that the ratio of the distances from $P$ to $A$ and $P$ to $B$ is $5:4$. This can be written as: $\frac{PA}{PB} = \frac{5}{4}$

Using the distance formula, we can express $PA$ and $PB$ as: $PA = \sqrt{(x-2)^2 + (y-1)^2}$ $PB = \sqrt{(x-1)^2 + (y-3)^2}$

Substituting these into the ratio equation: $\frac{\sqrt{(x-2)^2 + (y-1)^2}}{\sqrt{(x-1)^2 + (y-3)^2}} = \frac{5}{4}$

Step 3: Eliminate Square Roots and Simplify the Equation

To eliminate the square roots, we square both sides of the equation: $\left(\frac{\sqrt{(x-2)^2 + (y-1)^2}}{\sqrt{(x-1)^2 + (y-3)^2}}\right)^2 = \left(\frac{5}{4}\right)^2$ $\frac{(x-2)^2 + (y-1)^2}{(x-1)^2 + (y-3)^2} = \frac{25}{16}$

Expanding the squared terms: $(x-2)^2 + (y-1)^2 = x^2 - 4x + 4 + y^2 - 2y + 1 = x^2 + y^2 - 4x - 2y + 5$ $(x-1)^2 + (y-3)^2 = x^2 - 2x + 1 + y^2 - 6y + 9 = x^2 + y^2 - 2x - 6y + 10$

Substituting these back into the equation: $\frac{x^2 + y^2 - 4x - 2y + 5}{x^2 + y^2 - 2x - 6y + 10} = \frac{25}{16}$

Step 4: Cross-Multiply and Rearrange the Equation

Cross-multiplying: $16(x^2 + y^2 - 4x - 2y + 5) = 25(x^2 + y^2 - 2x - 6y + 10)$ $16x^2 + 16y^2 - 64x - 32y + 80 = 25x^2 + 25y^2 - 50x - 150y + 250$

Moving all terms to one side: $0 = 25x^2 - 16x^2 + 25y^2 - 16y^2 - 50x + 64x - 150y + 32y + 250 - 80$ $0 = 9x^2 + 9y^2 + 14x - 118y + 170$

The equation of the locus is: $9x^2 + 9y^2 + 14x - 118y + 170 = 0$

Step 5: Compare with the General Equation and Identify Coefficients

The general equation is given as $a x^2+b y^2+c x y+d x+e y+170=0$. Comparing this with our derived equation $9x^2 + 9y^2 + 14x - 118y + 170 = 0$, we can identify the coefficients: $a = 9$ $b = 9$ $c = 0$ $d = 14$ $e = -118$

Step 6: Calculate the Required Expression

We need to find the value of $a^2 + 2b + 3c + 4d + e$. Substituting the values of the coefficients: $a^2 + 2b + 3c + 4d + e = (9)^2 + 2(9) + 3(0) + 4(14) + (-118)$ $= 81 + 18 + 0 + 56 - 118$ $= 81 + 18 + 56 - 118$ $= 155 - 118$ $= 37$

Common Mistakes & Tips

• Remember to square both sides of the equation to eliminate square roots. Forgetting to square the ratio is a frequent error.
• Be careful with algebraic manipulations, especially when expanding squared terms and distributing negative signs.
• Double-check the signs of the coefficients when comparing the derived equation to the general form.

Summary

We found the locus of the point by using the distance formula and the given ratio. After simplifying the equation and comparing it to the general form, we identified the coefficients and calculated the value of the expression $a^2 + 2b + 3c + 4d + e$, which is 37.

The final answer is \boxed{37}, which corresponds to option (A).