The locus of a variable point whose distance | vedclass

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(A) Let the variable point be $P(x, y)$.According to the definition of a conic section,the distance from a fixed point (focus) is $e$ times the distan

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Ellipse

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(A) Let the variable point be $P(x, y)$.According to the definition of a conic section,the distance from a fixed point (focus) is $e$ times the distance from a fixed line (directrix).Here,the focus is $S(-2, 0)$,the directrix is $x = -\frac{9}{2}$,and the eccentricity $e = \frac{2}{3}$.Since $e Mathematically,$\sqrt{(x + 2)^2 + y^2} = \frac{2}{3} |x + \frac{9}{2}|$.Squaring both sides: $(x + 2)^2 + y^2 = \frac{4}{9} (x + \frac{9}{2})^2$.$(x^2 + 4x + 4) + y^2 = \frac{4}{9} (x^2 + 9x + \frac{81}{4})$.$x^2 + 4x + 4 + y^2 = \frac{4}{9}x^2 + 4x + 9$.$\frac{5}{9}x^2 + y^2 = 5$.Dividing by $5$: $\frac{x^2}{9} + \frac{y^2}{5} = 1$.This is the equation of an ellipse.

The locus of a variable point whose distance from $(-2, 0)$ is $\frac{2}{3}$ times its distance from the line $x = -\frac{9}{2}$ is

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