Question

V2
3
3. The ratio of the ordinates of a point and then point corresponding to it is
212
then eccentricity is (If P is a point
on an ellipse and PN is perpendicular to x-axis and if NP is produces to mete the auxiliary circle at Q.
Then Q is called the corresponding point of P)
2
VE
(a)
(b)
3
(b)
(c)
22
(d)
3
3​

Answer 1

The eccentricity of the ellipse is 1/√2

Step-by-step explanation:

From the property of ellipse

If Q is corresponding point to a point P on ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

And PN is perpendicular on the x-axis then PN will be the ordinate of the point P

Similarly if the corresponding point on the auxiliary circle is Q, the QN will be the ordinate of the corresponding point

It is evident that QN > PN

Given that

$\frac{PN}{QN}=\frac{1}{\sqrt{2}}$

From the property of ellipse we know that

$\frac{PN}{QN}=\frac{b}{a}$

Thus,

$\frac{b}{a}=\frac{1}{\sqrt{2}}$

Eccentricity of an ellipse is given by

$e=\sqrt{1-\frac{b^2}{a^2}}$     (here a > b)

$\implies e=\sqrt{1-(\frac{1}{\sqrt{2}})^2}$

$\implies e=\sqrt{1-\frac{1}{2}}$

$\implies e=\frac{1}{\sqrt{2}}$

Hope this answer is helpful.

Know More:

Q: find the eccentricity of an ellipse of which distance between the foci is 10 and that of focus and corresponding directrix is 15​ .

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