Standard Equation of Ellipse

Q.

The ellipse$E1:\frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point $(0,4)$ circumscribes the rectangle $R$. The eccentricity of the ellipse $E_2$ is

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

2. $\frac{\sqrt{3}}{2}$

3. $\frac{1}{2}$

4. $\frac{3}{4}$

Q.

The angle between the pair of lines $(x^2+y^2){\sin }^2\alpha ={(x\cos \theta -y\sin \theta )}^2$ is

1. $\theta$

2. $2\theta$

3. $\alpha$

4. $2\alpha$

Q. An ellipse, with foci at $(0,2)$ and $(0,–2)$ and minor axis of length $4$ units, passes through which of the following points?

1. $(2,\sqrt{2})$
2. $(1,2\sqrt{2})$
3. $(\sqrt{2},2)$
4. $(2,2\sqrt{2})$

Q. The locus of point of intersection of the lines
$\frac{tx}{a}-\frac{y}{b}+t=0,\frac{x}{a}+\frac{ty}{b}-1=0$ is $(t$ is a parameter and $a\ne b)$

1. parabola
2. circle
3. pair of straight lines
4. ellipse

Q. The length of the major axis and the minor axis of the ellipse $2x^2+3y^2-4x-12y+13=0$ are and respectively.

1. $\sqrt{2}$
2. $\sqrt{3}$
3. $\frac{2}{\sqrt{3}}$
4. $\frac{3}{\sqrt{2}}$

Q.

Let P be a variable point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with foci $F_1$ and $F_2$. If A is the area of the $\Delta P{F}_1{F}_2$, then the maximum value of A is

1. $b\sqrt{a^2-b^2}$

2. $b\sqrt{b^2-a^2}$
3. $a\sqrt{a^2-b^2}$
4. $a\sqrt{b^2-a^2}$

Q. The equation $\frac{x^2}{2-\lambda }+\frac{y^2}{\lambda -5}+1=0$ represents an ellipse, if

1. $\lambda \in (-\infty ,2)\cup (5,\infty )$
2. $\lambda \in (2,5)$
3. $\lambda \in (2,\frac{7}{2})\cup (\frac{7}{2},5)$
4. $\lambda \in (2,\frac{7}{2})\cup (5,\infty )$

Q. Let $S$ and $S^{\prime }$ be the two foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$ If the circle described on $S{S}^{\prime }$ as diameter touches the ellipse in real points, then $6e^2=$(where $e$ is eccentricity of ellipse)

Q. If $\frac{x^2}{3-\alpha }+\frac{y^2}{2}=1$ represents an ellipse whose major axis is along the $x-$axis, then the range of $\alpha$ is

1. $(1,\infty )$
2. $(-\infty ,1)$
3. $(-\infty ,-1)$
4. $(3,\infty )$

Q. The equation of the circle whose end points of the diameter are the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ is

1. $x^2+y^2=16$
2. $x^2+y^2=23$
3. $x^2+y^2=7$
4. $x^2+y^2=9$

Q. Let axes of ellipse be coordinate axes, $S$ and $S^{\prime }$ be foci, $B$ and $B^{\prime }$ are the endpoints of the minor axis. If $\sin \left(\angle SB{S}^{\prime }\right)=\frac{4}{5}$ and area of $SB{S}^{\prime }{B}^{\prime }$ is $20$ sq. unit, then the equation of ellipse is

1. $\frac{x^2}{20}+\frac{y^2}{16}=1$
2. $\frac{x^2}{18}+\frac{y^2}{13}=1$
3. $\frac{x^2}{25}+\frac{y^2}{16}=1$
4. $\frac{x^2}{25}+\frac{y^2}{20}=1$

Q. Equation of the circle passing through the focii of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and having centre at $(0,3)$ is

1. $x^2+y^2-6y-7=0$
2. $x^2+y^2-6y+7=0$
3. $x^2+y^2-6y-5=0$
4. $x^2+y^2-6y+5=0$

Q. If one of the foci of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ coincide with the focus of the parabola $y^2=8x$ and they intersect at a point where the ordinate is double the abscissa, then the value of $\left[b^2\right]$ is
(where [.] represents greatest integer function)

1. $5$
2. $19$
3. $10$
4. $6$

Q. For the ellipse $x^2+4y^2=9,$ which of the following option(s) is/are correct

1. the eccentricity is $\frac{1}{2}$
2. the length of latus-rectum is $\frac{3}{2}$
3. one focus is at $(3\sqrt{3},0)$
4. one directrix is $x=-2\sqrt{3}$

Q. The equation of an ellipse, centred at origin and passing through the points $(4,3)$ and $(-1,4),$ is

1. $7x^2+15y^2=247$
2. $49x^2+225y^2=105$
3. $7x^2+15y^2=105$
4. $7x^2+15y^2=147$

Q. Let $A$ and $B$ be two points on the major axis of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1,$ which are equidistant from the centre. If $C$ and $D$ are the images of these points in the line mirror $y=mx(m\ne 0),$ then the maximum area of quadrilateral $ACBD$ is

Q. The normal at $(4,4)$ to the parabola $y^2=4x$ is given by $L=0$. If $S:\frac{x^2}{36}+\frac{y^2}{20}-1=0$ be an ellipse, then

1. $L=0$ passes through a focus of the ellipse.
2. $L=0$ passes through a vertex of the ellipse.
3. $L=0$ parallel to the major axis of the ellipse.
4. $L=0$ parallel to the minor axis of the ellipse.