Standard Equation of Ellipse

Q. The equation of the ellipse, whose length of the major axis is $20$ and foci are $(0,\pm 5),$ is

1. $\frac{x^2}{10}+\frac{y^2}{5\sqrt{3}}=1$
2. $\frac{x^2}{25}+\frac{y^2}{100}=1$
3. $\frac{x^2}{75}+\frac{y^2}{100}=1$
4. $\frac{x^2}{10}+\frac{y^2}{25}=1$

Q. Let $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b.$ Let $E_2$ be another ellipse such that it touches the end points of major axis of $E_1$ and the foci of $E_2$ are the end points of minor axis of $E_1.$ If $E_1$ and $E_2$ have same eccentricities, then its value is

1. $\frac{-1+\sqrt{6}}{2}$
2. $\frac{-1+\sqrt{8}}{2}$
3. $\frac{-1+\sqrt{3}}{2}$
4. $\frac{-1+\sqrt{5}}{2}$

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

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. If a tangent of slope $\frac{1}{3}$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(0<b<a)$ is normal to the circle $x^2+y^2+2x+2y+1=0,$ then

1. the maximum value of $ab$ is $\frac{2}{3}$
2. $a\in (\sqrt{\frac{2}{5}},2)$
3. $a\in (\frac{2}{3},2)$
   
4. the maximum value of $ab$ is $1$

Q. If the tangent to the curve $y^2=x^3$ at $(m^2,m^3)$ is also a normal to the curve at $(M^2,M^3),$ then the value of $mM$ is

1. $-\frac{1}{3}$
2. $-\frac{4}{9}$
3. $-\frac{1}{9}$
4. $-\frac{2}{9}$

Q.

Find the equation of the ellipse whose foci are (4, 0) and (-4, 0), eccentricity = 1/3.

Q. If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $x\text{cosec}\alpha -y\sec \alpha =k\cot 2\alpha$ and $x\sin \alpha +y\cos \alpha =k\sin 2\alpha$ respectively, then $k^2$ is equal to :

1. $2p^2+q^2$
2. $p^2+4q^2$
3. $4p^2+q^2$
4. $p^2+2q^2$

Q. A circle has the same centre as an ellipse and passes through the foci $F_1$ & $F_2$ of the ellipse, such that the two curves intersect in $4$ points. Let $P$ be any one of their point of intersection such that the area of triangle $P{F}_1{F}_2$ is $30$ sq. unit. If the length of major axis of the ellipse is $17$ unit, then the distance between the foci is unit(s).

Q.

An ellipse is drawn by taking the diameter of the circle ${(x-1)}^2+y^2=1$as its semi-minor axis and the diameter of the circle$x^2+{(y-2)}^2=4$ as the semi-major axis, if the center of the ellipse at the origin and its axis are the coordinates axes, then the equation of the ellipse is

1. $4x^2+y^2=4$

2. $x^2+4y^2=8$

3. $4x^2+y^2=8$

4. $x^2+4y^2=16$

Q. Let a line $L$ pass through the point intersection of the lines $bx+10y–8=0\text{and}2x–3y=0,b\in R-\left\{\frac{4}{3}\right\}$. If the line $L$ also passes through the point $(1,1)$ and touches the circle $17(x^2+y^2)=16$, then the eccentricity of the ellipse $\frac{x^2}{5}+\frac{y^2}{5}=1\text{is}$

Q. The extremities of the latus rectum of an ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ is

1. $(\pm \sqrt{7},\pm \frac{9}{4})$
2. $(\pm \sqrt{9},\pm \frac{16}{5})$
3. $(\pm 4,\pm \frac{25}{4})$
4. $(\pm 6,\pm \frac{4}{5})$

Q. Find the equation of the ellipse in the standard form whose minor axis is equal to the distance between foci and whose latus-rectum is 10.

Q. The equation of the ellipse whose extremities of minor axis are $(3,1)$ and $(3,5)$ and eccentricity is $\frac{1}{2},$ is

1. $4x^2+3y^2-18x-24y+47=0$
2. $3x^2+4y^2+18x-24y-47=0$
3. $3x^2+4y^2-18x-24y+47=0$
4. $3x^2+4y^2-24x-18y+47=0$

Q.

Equation of the latus rectum of the ellipse $9x^2+4y^2-18x-8y-23=0$

1. $y=\pm \surd 5$

2. $y=-\surd 5$

3. $y=1\pm \surd 5$

4. $x=-1\pm \surd 5$

Q. If the length of the perpendicular drawn from the point $P(a,4,2),a>0$ on the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1}\text{is}2\sqrt{6}$ units and $Q({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the image of the point $P$ in this line, then $a+\sum _{i=1}^3{\alpha }_i$ is equal to :

1. $14$
2. $8$
3. $12$
4. $7$

Q. For the given ellipse $9x^2+25y^2-18x-100y-116=0$, which of the following is/are correct?

1. centre is $(1,2)$
2. eccentricity is $\frac{4}{5}$
3. end points of latus-rectum are $(5,\frac{19}{5})\text{and}(5,\frac{1}{5})$
4. length of major axis will be $10$ units

Q.

If a vector $\overrightarrow{r}$ has magnitude 14 and direction ratios 2, 3 and -6. Then, find the direction cosines and components of $\overrightarrow{r}$, given that $\overrightarrow{r}$ makes an acute angle with X- axis.

Q.

One of the points on the parabola $y^2=12x$ with focal distance $4$ is

1. $(3,6)$

2. $(9,6\sqrt{3})$

3. $(7,2\sqrt{21})$

4. $(8,4\sqrt{6})$

Q. The area of the triangle formed by the coordinate axes and a tangent to the curve $xy=a^2$ at the point $(x_1,y_1)$ on it is

1. $\frac{a^2{x}_1}{y_1}$
2. $\frac{a^2{y}_1}{x_1}$
3. $2a^2$
4. $4a^2$

Q.

If the line $x+2y+4=0$ cutting the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in points whose eccentric angles are ${30}^{\circ }$ and ${60}^{\circ }$ subtends a right angle at the origin then its equation is

1. none of these

2. $\frac{x^2}{8}+\frac{y^2}{4}=1$

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

4. $\frac{x^2}{4}+\frac{y^2}{16}=1$

Q. 94.A curve is governed by the equation y=cosx . Then what is the area enclosed by the curve and x -axis between x=0 and x=/2 is shaded region?

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. For the given ellipse $9x^2+25y^2-18x-100y-116=0$, which of the following is/are correct?

1. centre is $(1,2)$
2. eccentricity is $\frac{4}{5}$
3. end points of latus-rectum are $(5,\frac{19}{5})\text{and}(5,\frac{1}{5})$
4. length of major axis will be $10$ units

Q. Let $A,B$ be one of the ends of major and minor axes of an ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ respectively, where $A$ lies on positive $x-$ axis and $B$ lies on positive $y-$ axis. Let $L_1$ be a vertical line through $A$ and $L_2$ be a horizontal line through $B$, also $L_1$ and $L_2$ intersect at the point $P.$ If $S=0$ is a circle having $OP$ as radius, where $O$ is the origin and $P$ is the centre, then which of the following is/are correct?

1. $S=0$ intersects $x-$axis at $(8,0)$
2. $S=0$ intersects $y-$axis at $(0,\pm 5)$
3. $S=0$ intersects the ellipse at two points only.
4. $S=0$ does not intersects the ellipse.

Q. A ray emitting from the point $(-3,0)$ is incident on the ellipse $16x^2+25y^2=400$ at point $P$ with an ordinate $4.$ If the equation of the reflected ray after first reflection is $4x+3y=k,$ then the value of $k$ is

1. $12$
2. $-9$
3. $-12$
4. $9$

Q. An ellipse has its centre at (1, -1) and semi-major axis=8 and it passes through the point (1, 3). The equation of the ellipse is

1. $\frac{(x+1{)}^2}{64}+\frac{(y+1{)}^2}{16}=1$
   
2. $\frac{(x-1{)}^2}{64}+\frac{(y+1{)}^2}{16}=1$
   
3. $\frac{(x-1{)}^2}{16}+\frac{(y+1{)}^2}{64}=1$
   
4. $\frac{(x-1{)}^2}{64}+\frac{(y+1{)}^2}{16}=1$

Q. Using cofactors of elements of third column evaluate $\Delta =\left|\begin{array}{ccc}1& x& yz\\ 1& y& zx\\ 1& z& xy\end{array}\right|$

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. 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. $(3,\infty )$