Question

find the equation of the ellipse whose eccentricity is 1/2,one of the focii (2,3) and directix x=7. also find the length of the major and minor axis of the ellipse ​

Answer 1

Given:

Ellipse :

• Eccentricity ,e= 1/2
• Focii= (2,3)
• Directix, x= 7

To Find :

• Equation of the ellipse

${\purple{\boxed{\large{\bold{Ellipse}}}}}$

The ellipse is a conic section in which the eccentricity e is less than unity.

Solution :

Let P(x,y) be any point on the ellipse and PM be the perpendicular from P on the directix ,and let S(2,3) .

Then by definition ,

SP = e× PM

$\implies\sf\:SP^2=e^2\times\:PM^2$

$\implies\sf(x-2)^2+(y-3)^2=\dfrac{1}{2^2}\times\dfrac{(x-7)^2}{1^2}$

$\implies\sf4[(x-2)^2+(y-3)^2]=(x-7)^2$

$\implies\sf4(x^2-4x+4+y^2+9-6y]=x^2+49-14x$

$\implies\sf4x^2-16x+16+4y^2+36-24y=x^2+49-14x$

$\implies\sf3x^2-2x+3+4y^2-24y=0$

This is the equation of required Ellipse .

$\rule{200}2$

More About Ellipse :

➝Two Standard forms of the ellipse.

Standard equation $\tt\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ (a>b), where a and b are constants.and $\tt\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1$ (a>b).

Answer 2

To Find -

• Equation of ellipse.

Given -

• eccentricity = 1/2 .

• Focii = (2,3)

• Directrix = x-7 =0

Solution -

$\sf{ \implies \: \sqrt{(x - 2 {)}^{2} + (y - 3 {)}^{2} } = \frac{1}{2} }. | \frac{x - 7}{ \sqrt{ {1}^{2} } } |$

Squaring both sides -

$\sf{ \implies \: (x - 2 {)}^{2} + (y - 3 {)}^{2} = \frac{1}{4} \frac{(x - 7 {)}^{2} }{1 } }$

$\sf{ \implies \: (x - 2 {)}^{2} + (y - 3 {)}^{2} = \frac{1}{4} (\frac{ {x}^{2} + 49 - 14x}{1 }) }$

$\sf{ \implies \:4( (x - 2 {)}^{2} + (y - 3 {)}^{2} ) = {x}^{2} + 49 - 14x }$

$\sf{ \implies \:4 {x}^{2} + 4 {y}^{2} - 16x - 24y + 52 = {x}^{2} + 49 - 14x }$

$\underline{ \underline {\sf{ \implies \:3{x}^{2} + 4{y}^{2} - 2x - 24y + 3 = 0 }}}$

Ellipse -

An ellipse is the locus of a point on a plane which moves on the plane in such a way that the ratio of its distance from a fixed point which we can called as focus in the same plane to the distance from the fixed straight line which is also known as directrix is always constant and less than unity .