The set of real values of k for which the equation ( k + 1 )x^2 + 2( k - 1 )xy + y^2 - x + 2y + 3 = 0 represents an ellipse is :

For ellipse, ab - h^2 > 0 ( k + 1 ).1 - ( k - 1 )^2 > 0 or k( k - 3 ) < 0

the set of real values of k for which the equation k 1 x 2

The set of real values of $$k$$ for which the equation $$\left( {k + 1} \right){x^2} + 2\left( {k - 1} \right)xy + {y^2} - x + 2y + 3 = 0$$ represents an ellipse is :

A. $$\left( {0,\,3} \right)$$

B. $$\left( { - \infty ,\,0} \right)$$

C. $$\left( {3,\, + \infty } \right)$$

D. $$\left( { - \infty ,\,\infty } \right)$$

Answer : $$\left( {0,\,3} \right)$$

Solution :
For ellipse, $$ab - {h^2} > 0\,\,\, \Rightarrow \left( {k + 1} \right).1 - {\left( {k - 1} \right)^2} > 0{\text{ or }}k\left( {k - 3} \right) < 0$$

Releted MCQ Question on
Geometry >> Ellipse

Releted Question 1

Let $$E$$ be the ellipse $$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$$ and $$C$$ be the circle $${x^2} + {y^2} = 9.$$ Let $$P$$ and $$Q$$ be the points $$\left( {1,\,2} \right)$$ and $$\left( {2,\,1} \right)$$ respectively. Then-

A. $$Q$$ lies inside $$C$$ but outside $$E$$

B. $$Q$$ lies outside both $$C$$ and $$E$$

C. $$P$$ lies inside both $$C$$ and $$E$$

D. $$P$$ lies inside $$C$$ but outside $$E$$

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Releted Question 2

The radius of the circle passing through the foci of the ellipse $$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1,$$ and having its centre at $$\left( {0,\,3} \right)$$ is-

A. $$4$$

B. $$3$$

C. $$\sqrt {\frac{1}{2}} $$

D. $$\frac{7}{2}$$

View Answer

Releted Question 3

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse $$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1,$$ is-

A. $$\frac{{27}}{4}\,\,{\text{sq}}{\text{.}}\,{\text{units}}$$

B. $$9\,\,{\text{sq}}{\text{.}}\,{\text{units}}$$

C. $$\frac{{27}}{2}\,\,{\text{sq}}{\text{.}}\,{\text{units}}$$

D. $$27\,\,{\text{sq}}{\text{.}}\,{\text{units}}$$

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Releted Question 4

If tangents are drawn to the ellipse $${x^2} + 2{y^2} = 2,$$ then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is-

A. $$\frac{1}{{2{x^2}}} + \frac{1}{{4{y^2}}} = 1$$

B. $$\frac{1}{{4{x^2}}} + \frac{1}{{2{y^2}}} = 1$$

C. $$\frac{{{x^2}}}{2} + \frac{{{y^2}}}{4} = 1$$

D. $$\frac{{{x^2}}}{4} + \frac{{{y^2}}}{2} = 1$$

View Answer

The set of real values of $$k$$ for which the equation $$\left( {k + 1} \right){x^2} + 2\left( {k - 1} \right)xy + {y^2} - x + 2y + 3 = 0$$ represents an ellipse is :

Releted MCQ Question on
Geometry >> Ellipse

Let $$E$$ be the ellipse $$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$$ and $$C$$ be the circle $${x^2} + {y^2} = 9.$$ Let $$P$$ and $$Q$$ be the points $$\left( {1,\,2} \right)$$ and $$\left( {2,\,1} \right)$$ respectively. Then-

The radius of the circle passing through the foci of the ellipse $$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1,$$ and having its centre at $$\left( {0,\,3} \right)$$ is-

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse $$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1,$$ is-

If tangents are drawn to the ellipse $${x^2} + 2{y^2} = 2,$$ then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is-

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Ellipse

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The set of real values of $$k$$ for which the equation $$\left( {k + 1} \right){x^2} + 2\left( {k - 1} \right)xy + {y^2} - x + 2y + 3 = 0$$ represents an ellipse is :

Releted MCQ Question on Geometry >> Ellipse

Let $$E$$ be the ellipse $$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$$ and $$C$$ be the circle $${x^2} + {y^2} = 9.$$ Let $$P$$ and $$Q$$ be the points $$\left( {1,\,2} \right)$$ and $$\left( {2,\,1} \right)$$ respectively. Then-

The radius of the circle passing through the foci of the ellipse $$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1,$$ and having its centre at $$\left( {0,\,3} \right)$$ is-

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse $$\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1,$$ is-

If tangents are drawn to the ellipse $${x^2} + 2{y^2} = 2,$$ then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is-

Practice More Releted MCQ Question on Ellipse

Practice More MCQ Question on Maths Section

Releted MCQ Question on
Geometry >> Ellipse

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Ellipse