Definition of Ellipse

Q.

Let a vector $\begin{array}{l}\alpha \hat{i}+\beta \end{array}\hat{j}$ be obtained by rotating the vector $\begin{array}{l}\sqrt{3}\hat{i}+\hat{j}\end{array}$by an angle $45°$ about the origin in counter clockwise direction in the first quadrant. Then the area of triangle having vertices $(ɑ,\beta ),(0,\beta )$ and $(0,0)$ is equal to:

1. $1$

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

3. $\sqrt{2}$

4. $2\sqrt{2}$

Q. If the latus rectum of an ellipse is equal to half of its minor axis, then its eccentricity is

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

Q.

A hyperbola having the transverse axis of length $\sqrt{2}$ has the same foci as that of the ellipse $3x^2+4y^2=12$, then this hyperbola does not pass through which of the following points -

1. $\left(\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right)$

2. $\left(1,\frac{-1}{\sqrt{2}}\right)$

3. $\left(-\sqrt{\frac{3}{2}},1\right)$

4. $\left(\frac{1}{\sqrt{2}},0\right)$

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.

If three points A, B&C have position vectors (1, x, 3), (3, 4, 7) and (y, -2, -5) respectively.If they are collinear find the values of x&y.

Q. If the length of the latus rectum of an ellipse is $4$ units and the distance between a focus and its nearest vertex on the major axis is $\frac{3}{2}$ units, then its eccentricity is

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

Q. The slope of a line passing through P$(2,3)$ and intersecting the line, $x+y=7$ at a distance of 4 units from P, is:

1. $\frac{1-\sqrt{7}}{1+\sqrt{7}}$
2. $\frac{\sqrt{5}-1}{\sqrt{5}+1}$
3. $\frac{1-\sqrt{5}}{1+\sqrt{5}}$
4. $\frac{\sqrt{7}-1}{\sqrt{7}+1}$

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 $S$ 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. The eccentricity of an ellipse $9x^2+16y^2=144$ is

1. $\frac{\sqrt{7}}{4}$
2. $\frac{2}{5}$
3. $\frac{\sqrt{3}}{5}$
4. $\frac{\sqrt{5}}{3}$

Q. From any point $P$ lying in the first quadrant on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1,PN$ is drawn perpendicular to the major axis and produced at $Q$ so that $NQ$ equals to $P{S}^{\prime },$ where $S^{\prime }$ can be any foci. Then the locus of $Q$ can be

1. $3x-5y+25=0$
2. $3x+5y+25=0$
3. $3x+5y=25$
4. $3x-5y=25$

Q.

If the distance between the foci of an ellipse is $6$ and the distance between its directrices is $12$, then the length of its latus rectum is

1. $4\sqrt{3}$

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

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

4. $3\sqrt{2}$

Q.

Why is the eccentricity of an ellipse between $0$ and $1$?

Q. If a straight line through the point $P(\lambda ,2)$ where $\lambda \ne 0$, meets the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ at $A$ and $D$ and meets the coordinate axis at $B$ and $C$ such that $PA\cdot PD=PB\cdot PC,$ then range of $\lambda$ is
(correct answer + 2, wrong answer - 0.50)

1. $[12,\infty )$
2. $(-\infty ,-12]\cup [12,\infty )$
3. $(-\infty ,-6]\cup [6,\infty )$
4. $[6,\infty )$

Q.

$C_{1}and{C}_2$ are circles of unit radius with centres at (0, 0) and (1, 0) respectively. $C_3$ is a circle of unit radius passes through the centres of the circles $C_{1}and{C}_2$ and have its centre above x-axis. Equation of the common tangent to $C_{1}and{C}_3$ which does not pass through $C_2$ is

1. 

2. 

3. 

4. 

Q. If the length of the major axis of the ellipse $(5x-10{)}^2+(5y+15{)}^2=\frac{(3x-4y+7{)}^2}{4},$ is $k$ units, then $3k=$

Q. Consider two straight lines, each of which is tangent to both the circle $x^2+y^2=\frac{1}{2}$ and the parabola $y^2=4x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement(s) is (are) TRUE?

1. For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$
2. For the ellipse, the eccentricity is $\frac{1}{2}$
3. For the ellipse, the length of the latus rectum is $1$ unit
4. For the ellipse, the length of the latus rectum is $\frac{1}{2}$

Q. Equation of the ellipse with focus $(3,-2)$,
eccentricity $\frac{3}{4}$ and directrix $2x-y+3=0$ is

1. $44x^2+36xy+71y^2-588x+374y+959=0$
2. $44x^2+36xy+71y^2-374x-528y+756=0$
3. $44x^2+36xy+71y^2-135x-47y+859=0$
4. $44x^2+36xy+71y^2-125x-274y+659=0$

Q. An underpass bridge is in the shape of a semi ellipse. It is $400$ m long and has a maximum depth $10$ m at the middle point. The depth of the bridge at a point which is at a distance of $80$ m form one end is m

Q.

Let $A(acos\theta ,bsin\theta )$ is a variable point, $S=(ap,0)$ and $S^{\prime }\equiv (-ap,0)$ are two fixed points where $p=\sqrt{\frac{a^2-b^2}{a^2}}$. If the locus of Incentre of triangle ASS' is a conic then the eccentricity of the conic in terms of p is

1. $\sqrt{\frac{p}{1+p}}$
2. $\sqrt{\frac{1-p}{1+p}}$
3. $\frac{p}{2(1+p)}$
4. $\sqrt{\frac{2p}{1+p}}$

Q. A ladder $12$ units long slides in a vertical plane with its ends in contact with a vertical wall and a horizontal floor along $x-$axis. Then the locus of a point on the ladder $4$ units from its foot, is

1. $\frac{x^2}{64}+\frac{y^2}{32}=1$
2. $\frac{x^2}{8}+\frac{y^2}{4}=1$
3. $\frac{x^2}{64}+\frac{y^2}{16}=1$
4. $\frac{x^2}{32}+\frac{y^2}{32}=1$

Q. Let $S_1$ and $S_2$ are the unit circles with centres at $C_1(0,0)$ and $C_2(1,0)$ respectively. Let $S_3$ is another circle of unit radius, passes through $C_1$ and $C_2$ and its centre is above the $x$ -axis. If equation of common tangent to $S_1$ and $S_3,$ which does not cut $S_2,$ is $ax+by+2=0$ then find the value of $(a^2-b)$.

Q. The number of the points where the function f(x)=min{|x|-1, |x-2|-1} is NOT derivable

Q. $\mathrm{If}P(x,y)\mathrm{is}\mathrm{any}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{ellipse}$$16x^2+25y^2=400$$\mathrm{and}$$F_1=(3,0),F_2=(-3,0),\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}{\mathrm{PF}}_1+{\mathrm{PF}}_2\mathrm{is}$

1. 2
2. 5
3. 10
4. 20

Q.

Be the position of the point (-3, -2) with respect to the circle whose equation is x2+ y2-3x+2y-19=0

Q. the number of points where f(x) = |x^2 - 3x + 2| is non differentiable is

Q. Find the equation of ellipse whose focus is $(1,-2)$, the directrix $3x-2y+5=0$ and eccentricity equal to $\frac{1}{2}$.

Q. Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ where $y_1,y_2<0,$ be the end points of the latus rectum of the ellipse $x^2+4y^2=4.$ Then equation(s) of the parabola with latus rectum $PQ$ is/are

1. $x^2+2\sqrt{3}y=3+\sqrt{3}$
2. $x^2-2\sqrt{3}y=3+\sqrt{3}$
3. $x^2+2\sqrt{3}y=3-\sqrt{3}$
4. $x^2-2\sqrt{3}y=3-\sqrt{3}$

Q. Find the equation of ellipse when: Focus is $(0,1)$, directrix is $x+y=0$ and $e=\frac{1}{2}$

Q. The eccentricity of the ellipse $25x^2+16y^2=400$ is

1. $\frac{3}{5}$
2. $\frac{1}{3}$
3. $\frac{1}{5}$
4. $\frac{2}{5}$