Find the locus of a point which moves in such a way that the sum of its distances from(4,3) and (4,1) is 5.

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Solution

Let the moving or variable point be $P (x, y)$ and the given points be $A (4, 3)$ and $B (4, 1)$
By the conditions of the problem,
$P A + P B = 5$
$P A = 5 - P B$
$\Rightarrow \sqrt{{(x - 4)}^{2} + {(y - 3)}^{2}} = 5 - \sqrt{{(x - 4)}^{2} + {(y - 1)}^{2}}$
$\Rightarrow {(x - 4)}^{2} + {(y - 3)}^{2} = 25 + {(x - 4)}^{2} + {(y - 1)}^{2} - 10 \sqrt{{(x - 4)}^{2} + {(y - 1)}^{2}}$
$\Rightarrow y^{2} - 6 y + 9 = 25 + y^{2} - 2 y + 1 - 10 \sqrt{{(x - 4)}^{2} + {(y - 1)}^{2}}$
$\Rightarrow 10 \sqrt{{(x - 4)}^{2} + {(y - 1)}^{2}} = 4 y + 17$
Squaring both sides again
$100 (x^{2} - 8 x + 16 + y^{2} - 2 y + 1) = {(4 y + 17)}^{2}$
$\Rightarrow 100 (x^{2} + y^{2} - 8 x - 2 y + 17) = 16 y^{2} + 136 y + 284$
$\Rightarrow {100 x}^{2} + {100 y}^{2} - {16 y}^{2} - 800 x - 200 y - 136 y + 1700 - 284 = 0$
${100 x}^{2} + 84 y^{2} - 800 x - 336 y + 1411 = 0$

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