(D) By the definition of an ellipse,the distance from any point $P(x, y)$ on the ellipse to the focus $S(0, 0)$ is $e$ times the distance to the direc

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(D) By the definition of an ellipse,the distance from any point $P(x, y)$ on the ellipse to the focus $S(0, 0)$ is $e$ times the distance to the directrix $x = 4$.$SP^2 = e^2 \times (\text{distance to directrix})^2$$x^2 + y^2 = (\frac{1}{2})^2 (x - 4)^2$$x^2 + y^2 = \frac{1}{4} (x^2 - 8x + 16)$$4x^2 + 4y^2 = x^2 - 8x + 16$$3x^2 + 8x + 4y^2 = 16$Multiply by $3$ to complete the square for $x$:$9x^2 + 24x + 12y^2 = 48$$(3x + 4)^2 - 16 + 12y^2 = 48$$(3x + 4)^2 + 12y^2 = 64$

Class 11 Mathematics 10-2. Parabola, Ellipse, Hyperbola Ellipse

Medium

$A$ focus of an ellipse having eccentricity $e = \frac{1}{2}$ is at $(0,0)$ and a directrix is the line $x = 4$. Then the equation of one such ellipse is

TS EAMCET 2021 All TS EAMCET

A
$\frac{9x^2}{64} + \frac{3y^2}{16} = 1$
B
$\frac{(2x+1)^2}{32} + \frac{y^2}{16} = 1$
C
$\frac{(3x+4)^2}{64} + \frac{y^2}{32} = 1$
D
$(3x+4)^2 + 12y^2 = 64$

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10-2. Parabola, Ellipse, Hyperbola

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$A$ focus of an ellipse having eccentricity $e = \frac{1}{2}$ is at $(0,0)$ and a directrix is the line $x = 4$. Then the equation of one such ellipse is

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