A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16. the locus of the point is
(a) 3x² + 4y² = 192
(b) 4x² + 3y² = 192
(c) x² + y² = 12
(d) none of these

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Solution

Let us suppose the point is P(x, y).
According to given condition
$\sqrt{{(x - 4)}^{2} + {(y - 0)}^{2}} = \frac{1}{2} |\frac{x - 16}{\sqrt{1^{2} + 0^{2}}}| i . e . \sqrt{{(x - 4)}^{2} + y^{2}} = \frac{1}{2} |x - 16| i . e . 4 [{(x - 4)}^{2} + y^{2}] = {(x - 16)}^{2} i . e . 4 [{(x - 4)}^{2} + y^{2}] = {(x - 16)}^{2} i . e . 4 [x^{2} + 16 - 8 x + y^{2}] = x^{2} + {(16)}^{2} - 32 x i . e . 4 x^{2} + 64 - 32 x + 4 y^{2} = x^{2} + {(16)}^{2} - 32 x i . e . 3 x^{2} + 4 y^{2} = 192$

Hence, the correct answer is option A.

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