Home
Class 12
MATHS
A parabola is drawn with focus at one of...

A parabola is drawn with focus at one of the foci of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` . If the latus rectum of the ellipse and that of the parabola are same, then the eccentricity of the ellipse is (a) `1-1/(sqrt(2))` (b) `2sqrt(2)-2` (c) `sqrt(2)-1` (d) none of these

A

`1-(1)/(sqrt(2))`

B

`2sqrt(2)-2`

C

`sqrt(2)-1`

D

none of these

Text Solution

Verified by Experts

(3) The equation of the ellipse is `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
The equation of the parabola with S)ae,0) and directrix x+ae=o is `y^(2)=4aex`.
Now, the length of latus rectum of the ellipse is `2b^(2)//a` and that of the parabola is 4ae.

For the latus recta tot be equal , we get
`(2b^(2))/(a=4ae`
or `(2a^(2)(1-e^(2)))/(a)=4ae`
`or 1-e^(2)=2e`
or `e^(2)2e-1=0`
Therefore, `e=(-2+-sqrt(8))/(2)=+-sqrt(2)`
Hence, `e=sqrt(2)-1`
Promotional Banner

Similar Questions

Explore conceptually related problems

If the eccentricity of the ellipse, x^2/(a^2+1)+y^2/(a^2+2)=1 is 1/sqrt6 then latus rectum of ellipse is

Pa n dQ are the foci of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and B is an end of the minor axis. If P B Q is an equilateral triangle, then the eccentricity of the ellipse is 1/(sqrt(2)) (b) 1/3 (d) 1/2 (d) (sqrt(3))/2

Pa n dQ are the foci of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and B is an end of the minor axis. If P B Q is an equilateral triangle, then the eccentricity of the ellipse is 1/(sqrt(2)) (b) 1/3 (d) 1/2 (d) (sqrt(3))/2

Find the equation of the normal to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 at the positive end of the latus rectum.

Find the eccentricity of an ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 whose latus rectum is half of its major axis.

If the normal at one end of the latus rectum of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 passes through one end of the minor axis, then prove that eccentricity is constant.

If the normal at one end of the latus rectum of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 passes through one end of the monor axis, then prove that eccentricity is constant.

The ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 is such that it has the least area but contains the circle (x-1)^(2)+y^(2)=1 The eccentricity of the ellipse is

If the ellipse x^2/a^2+y^2/b^2=1 (b > a) and the parabola y^2 = 4ax cut at right angles, then eccentricity of the ellipse is (a) (3)/(5) (b) (2)/(3) (c) (1)/(sqrt(2)) (d) (1)/(2)

There are exactly two points on the ellipse (x^2)/(a^2)+(y^2)/(b^2) =1 whose distance from the centre of the ellipse are equal to sqrt((3a^2-b^2)/(3)) . Eccentricity of this ellipse is