A point moves so that the square of its distance from the point (3, –2) is numerically equal to its distance from the line 5x – 12y = 3. The equation of its locus is __________.

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Solution

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Let P(h, k) be the given point.
Such that square of the distance from the point (3, −2) is equal to the distance from the line 5x − 12y = 3
$i . e . {(\sqrt{{(h - 3)}^{2} + {(k + 2)}^{2}})}^{2} = \frac{|5 h - 12 k - 3|}{\sqrt{5^{2} + 12^{2}}} i . e . {(h - 3)}^{2} + {(k + 2)}^{2} = \frac{|5 h - 12 k - 3|}{13} i . e . h^{2} + 9 - 6 h + k^{2} + 4 + 4 k = \frac{|5 h - 12 k - 3|}{13} i . e . h^{2} + k^{2} - 6 h + 4 k + 13 = \frac{|5 h - 12 k - 3|}{13} i . e . 13 (h^{2} + k^{2}) - 78 h + 52 k + 169 = |5 h - 12 k - 3| i . e . 13 (h^{2} + k^{2}) - 78 h - 5 h + 52 k + 12 k + 172 = 0 i . e . 13 (h^{2} + k^{2}) - 83 h + 64 k + 172 = 0$

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