(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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Ellipse

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If the curves $\frac{x^{2}}{a}+\frac{y^{2}}{b}=1$ and $\frac{x^{2}}{c}+\frac{y^{2}}{d}=1$ intersect each other at an angle of $90^{\circ}$,then which of the following relations is $TRUE$?

MediumJEE MAIN 2021

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Let $L$ be a common tangent line to the curves $4x^{2} + 9y^{2} = 36$ and $(2x)^{2} + (2y)^{2} = 31$. Then the square of the slope of the line $L$ is ..... .

DifficultJEE MAIN 2021

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Let the point $L$ lying in the first quadrant be one end of a latus rectum of the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$. Let $P$ and $Q$ be the points where the normal drawn at $L$ to this given ellipse meets the major axis and the minor axis. Then the distance between $P$ and $Q$ is

DifficultAP EAMCET 2023

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Find the equation for the ellipse that satisfies the given conditions: Length of major axis $= 26$,foci $= (\pm 5, 0)$.

Medium

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Find the equation for the ellipse that satisfies the given conditions: $b=3, c=4,$ centre at the origin; foci on the $x$-axis.

Medium

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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

Similar Questions

If the curves $\frac{x^{2}}{a}+\frac{y^{2}}{b}=1$ and $\frac{x^{2}}{c}+\frac{y^{2}}{d}=1$ intersect each other at an angle of $90^{\circ}$,then which of the following relations is $TRUE$?

Let $L$ be a common tangent line to the curves $4x^{2} + 9y^{2} = 36$ and $(2x)^{2} + (2y)^{2} = 31$. Then the square of the slope of the line $L$ is ..... .

Let the point $L$ lying in the first quadrant be one end of a latus rectum of the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$. Let $P$ and $Q$ be the points where the normal drawn at $L$ to this given ellipse meets the major axis and the minor axis. Then the distance between $P$ and $Q$ is

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