Tag: equation of ellipse

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Questions Related to equation of ellipse

Multiple choice maths ellipse special cases of an ellipse eccentricity equation of ellipse

If circle whose diameter is major axis of ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ meets minor axis at point P and orthocentre of $\Delta PF _{1}F _{2}$ lies on ellipse where $F _{1}$  and $F _{2}$ are foci of ellipse, then square of eccentricity of ellipse, is 

  1. $2 sin\frac{\pi }{10}$
  2. $2 sin\frac{\pi }{12}$
  3. $2 sin\frac{\pi }{4}$
  4. $2 sin\frac{\pi }{2}$
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Explanation

The problem involves geometric properties of an ellipse. Solving for the orthocenter condition leads to the relation involving the eccentricity squared.

Multiple choice maths ellipse special cases of an ellipse eccentricity equation of ellipse

An ellipse has foci (3, 1), (1, 1) and it passes through point (1, 3). Its eccentricity is equal to 

  1. $\sqrt { 2 } -1$
  2. $\sqrt { 3 } -1$
  3. $\cfrac { 1 }{ 2 } $
  4. $\cfrac { 1 }{ 3 } $
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Explanation

Given foci (3,1) and (1,1), the center is (2,1) and 2ae = 2, so ae = 1. Using the point (1,3), the sum of distances to the foci equals 2a.

Multiple choice maths ellipse special cases of an ellipse eccentricity equation of ellipse

The ellipse $E _1:\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$ is inscribed in a rectangle R whose sides are parallel to the coordinates axis. Another ellipse $E _2$ passing through the point $(0, 4)$ circumscribes the rectangle R. The eccentricity of the ellipse $E _2$ is?

  1. $\dfrac{\sqrt{2}}{2}$
  2. $\dfrac{\sqrt{3}}{2}$
  3. $\dfrac{1}{2}$
  4. $\dfrac{3}{4}$
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Multiple choice maths ellipse special cases of an ellipse eccentricity equation of ellipse

Eccentricity of the ellipse $5x^{2}+6xy+5y^{2}=8$ is

  1. $\dfrac {1}{\sqrt {2}}$
  2. $\dfrac {\sqrt {3}}{2}$
  3. $\sqrt {\dfrac {2}{3}}$
  4. $\dfrac {1}{\sqrt {3}}$
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Explanation

For 5x^2 + 6xy + 5y^2 = 8, rotate the axes to eliminate the xy term. The eigenvalues of the matrix determine the semi-axes, leading to e = 1/sqrt(2).

Multiple choice maths ellipse special cases of an ellipse eccentricity equation of ellipse

An ellipse has $OB$ as its semi-minor axis. $F _{1}$ and $F _{2}$ are its foci and angle $F _{1}BF _{2}$ is a right angle. The eccentricity of the ellipse is 

  1. $1/\sqrt{2}$
  2. $1/2$
  3. $1/\sqrt{3}$
  4. $2/\sqrt{3}$
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Explanation

If angle F1BF2 = 90 degrees, then in the right triangle F1OB, OB = OF1 = ae. Since b = ae, and b^2 = a^2(1-e^2), we get a^2e^2 = a^2(1-e^2), so 2e^2 = 1.

Multiple choice maths ellipse special cases of an ellipse eccentricity equation of ellipse

The tangent at any point $P\left(a\cos\theta,b\sin\theta\right)$ on the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ meets the auxiliary circle at two points which subtend a right angle at the center ,then eccentricity is 

  1. $\dfrac{1}{\sqrt{1+\sin^{2}\theta}}$
  2. $\dfrac{1}{\sqrt{2-\cos^{2}\theta}}$
  3. $\dfrac{1}{\sqrt{1+\tan^{2}\theta}}$
  4. $none\ of\ these$
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Explanation

The condition that the tangent meets the auxiliary circle at points subtending 90 degrees at the center relates the coordinates to the eccentricity.