The equation of the locus of the point whose distances from the point P and the line AB are equal, is [2010]

Question

Tangents are drawn from the point P(3, 4) to the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ touching the ellipse at points A and B. [2010]

Q. The equation of the locus of the point whose distances from the point P and the line AB are equal, is

Smart Achievers · Accepted Answer

(1)

$Clearly the moving point traces a parabola with focus at P (3, 4) and directrix as$

$A B : \frac{y - 0}{x - 3} = - \frac{1}{3} \Rightarrow x + 3 y - 3 = 0$

$∴$ $Equation of parabola is$

${(x - 3)}^{2} + {(y - 4)}^{2} = \frac{{(x + 3 y - 3)}^{2}}{10}$

$\Rightarrow 9 x^{2} + y^{2} - 6 x y - 54 x - 62 y + 241 = 0$

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