Question:
<p>If the locus of the point, whose distances from the point <span class="math-tex">\((2,1)\)</span> and <span class="math-tex">\((1,3)\)</span> are in the ratio <span class="math-tex">\(5: 4\)</span>, is <span class="math-tex">\(a x^{2}+b y^{2}+\)</span><span class="math-tex">\(c x y+d x+e y+170=0\)</span>, then the value of <span class="math-tex">\(a^{2}+2 b+3 c+4 d+e\)</span> is equal to:</p>
<p style="display:inline">437</p>
<p style="display:inline">5</p>
<p style="display:inline">-27</p>
<p style="display:inline">37</p>
Step-by-Step Solution
Key Concept: The locus of a point with a constant ratio of distances from two fixed points is derived by applying the distance formula and squaring the ratio to form a quadratic equation in x and y.
<p>Let <span class="math-tex">\(P(x, y)\)</span><br />
<span class="math-tex">\(\frac{(x-2)^{2}+(y-1)^{2}}{(x-1)^{2}+(y-3)^{2}}=\frac{25}{16}\)</span><br />
<span class="math-tex">\(\Rightarrow 9 x^{2}+9 y^{2}+\)</span><span class="math-tex">\(14 x-118 y+170=0\)</span><br />
Now, <span class="math-tex">\(a^{2}+2 b+3 c+45+e\)</span><br />
<span class="math-tex">\(=81+18+0+56-118\)</span><br />
<span class="math-tex">\(=155-118=37\)</span></p>
Correct Answer: D