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Standard XII

Mathematics

Centre of Ellipse

The ellipse ...

Question

The ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ cuts x axis at A and y axis at B and the line joining the focus S and B makes an angle $\frac{3 π}{4}$ with x-axis. Then the eccentricity of the ellipse is

\frac{1}{\sqrt{2}}

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\frac{1}{2}

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\frac{\sqrt{3}}{2}

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\frac{1}{3}

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Solution

The correct option is D $\frac{1}{\sqrt{2}}$
REF.Image
B is (0,b) S is (ae,0)
$\frac{- b}{a e} = t a n \frac{3 π}{4}$
$e = \frac{b}{a}$
$\frac{b}{a} = \sqrt{1 - \frac{b^{2}}{a^{2}}}$ $\frac{b}{a} = \frac{1}{\sqrt{2}}$
$e = \frac{1}{\sqrt{2}}$
A is correct

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Similar questions

Q. The ellipse cuts

x

axis at

A

and

y

axis at

B

and the line joining the focus

S

and

B

makes an angle

\frac{3 π}{4}

with

x

-axis, then the eccentricity of the ellipse is:

Q. The normal at any point

P

on the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b)

meets the x-axis at

A

and y-axis at

B

. If

P A : P B = 1 : 4,

then the eccentricity of the ellipse is

Q. If the chord joining two points whose eccentric angle are

α a n d β

. Cuts the major axis of the ellipse

\frac{x^{2}}{a^{2}} + y^{2} b^{2}

= 1 at a distance c from the centre, then

c o t \frac{α}{2}, c o t \frac{β}{2}

can be equal to

Q. If a chord joining two points whose eccentric angles are

α, β

cut the major axis of the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

, at a distance

d

from the center, then

tan \frac{α}{2} tan \frac{β}{2} =

Q. If the points of intersection of the circle

x^{2} + y^{2} = 16

with

x - axis

are foci of an ellipse and points of intersection of circle

x^{2} + y^{2} = 16

with

y - axis

are end points of minor axis of the same ellipse, then eccentricity of the ellipse is

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The ellipse x2a2+y2b2=1 cuts x axis at A and y axis at B and the line joining the focus S and B makes an angle 3π4 with x-axis. Then the eccentricity of the ellipse is

The correct option is D 1√2REF.ImageB is (0,b) S is (ae,0)−bae=tan3π4e=baba=√1−b2a2 ba=1√2e=1√2A is correct

The ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ cuts x axis at A and y axis at B and the line joining the focus S and B makes an angle $\frac{3 π}{4}$ with x-axis. Then the eccentricity of the ellipse is

The correct option is D $\frac{1}{\sqrt{2}}$
REF.Image
B is (0,b) S is (ae,0)
$\frac{- b}{a e} = t a n \frac{3 π}{4}$
$e = \frac{b}{a}$
$\frac{b}{a} = \sqrt{1 - \frac{b^{2}}{a^{2}}}$ $\frac{b}{a} = \frac{1}{\sqrt{2}}$
$e = \frac{1}{\sqrt{2}}$
A is correct