Focii of Ellipse

Q.

The distance between the foci of the ellipse $3x^2+4y^2=48$ is

1. $2$

2. $6$

3. $8$

4. $4$

Q. The equation of the ellipse having its centre at the point $(2,-3)$, one focus at $(3,-3)$ and one vertex at $(4,-3)$ is

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

Q. For the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, which of the following is/are true?

1. eccentricity is $\frac{3}{5}$
2. Length of minor axis is $8$ units
3. Length of major axis $10$ units
4. Centre of ellipse is $(0,0)$

Q. Let $C$ be a circle passing through the points $A(2,-1)$ and $B(3,4)$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle $(x-5{)}^2+(y-1{)}^2=\frac{13}{2}$, then $r^2$ is equal to :

Q. The coordinates of the foci of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ is/are

1. $(\sqrt{5},0)$
2. $(2\sqrt{5},0)$
3. $(-\sqrt{5},0)$
4. $(-2\sqrt{5},0)$

Q. An equilateral triangle is inscribed in an ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1)$, one altitude is contained in the $y-$axis, and the length of each side is $\sqrt{\frac{m}{n}},$ where $m$ and $n$ are relatively prime positive integers. Then the value of $(m+n)$ is

Q.

The three normals from a point to the parabola $y^2=4ax$ cut the axes in points, whose distances from the vertex are in A.P., then the locus of the point is

1. $27a{y}^2=2(x-2a{)}^3$
2. $27a{y}^3=2(x-2a{)}^2$
3. $9a{y}^2=2(x-2a{)}^3$
4. $9a{y}^3=2(x-2a{)}^2$

Q. The equation of the circle with centre at (1, -2) and passing through the centre of the given circle $x^2+y^2+2y-3=0,$ is

1. 
2. 
3. 
4. 

Q. An ellipse passing through origin has its foci at (5, 12) and (24, 7). Then its eccentricity is

1. 
2. 
3. 
4. 

Q. A line passing through $P(6,4)$ meets the coordinates axes at $A$ and $B$ respectively. If $O$ is the origin, then the locus of the centre of the circumcircle of triangle $OAB$ is

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

Q. If $(3,4)$ and $(8,6)$ are the foci of an ellipse passing through the origin, then the eccentricity of the ellipse is

1. $\frac{1}{\sqrt{2}}$
2. $\frac{\sqrt{29}}{15}$
3. $\frac{1}{2}$
4. $\sqrt{\frac{15}{29}}$

Q.

A$(x_1,y_1)$ and B$(x_2,y_2)$ are the end-points of diameter of a circle. Find the equation of the circle.

1. (x + x₁) (x + x₂) + (y + y₁) (y + y₂) = 0

2. (x - x₁) (x - x₂) + (y - y₁) (y - y₂) = 0

3. (x - x₁) (x - x₂) + (y + y₁) (y + y₂) = 0

4. (x + x₁) (x + x₂) - (y + y₁) (y + y₂) = 0

Q. The perimeter of a triangle is 20 and the points (-2, -3) and (-2, 3) are two of the vertices of it. Then the locus of third vertex is :

1. 
2. 
3. 
4. 

Q. The graph of $\sqrt{x}+\sqrt{y}=\sqrt{2}$ is

1. 
2. 
3. 
4. 

Q. A point $P$ moves such that three mutually perpendicular lines $PA,PB$ and $PC$ are drawn from it cutting $x,y$ and $z$ axis at $A,B\text{and}C$ respectively. The volume of tetrahedron $OABC$ is $\frac{4}{3}$ cubic units (where $O$ is the origin). If locus of $P$ is $(x^2+y^2+z^2{)}^{\mu }=(\lambda xyz)$ then which of the following is correct $(\lambda ,\mu \in R)$?

1. $\lambda -\mu =61$
2. $\lambda +\mu =67$
3. $\lambda \mu =192$
4. $(\lambda {)}^{\frac{1}{\mu }}=4$

Q. For the horizontal ellipse, the difference between the length of the major axis and the latus rectum of an ellipse is

1. $ae$
2. $2ae$
3. $a{e}^2$
4. $2a{e}^2$

Q. The point $P$ on the ellipse $4x^2+9y^2=36$ is such that the area of the $△P{F}_1{F}_2=\sqrt{10}\text{sq. units}$, where $F_1,F_2$ are foci. Then the coordinates of $P$ will be

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

Q. The point $P$ on the ellipse $4x^2+9y^2=36$ is such that the area of the $△P{F}_1{F}_2=\sqrt{10}\text{sq. units}$, where $F_1,F_2$ are foci. Then the coordinates of $P$ will be

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

Q.

If $(–4,3)$ and$(4,3)$ are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the interior of the triangle.

Q.

let $S$, $S^1$ be the foci of an ellipse. If $\angle BS{S}^1$ = $\theta$, where B is any point on the ellipse. Then its eccentricity is

1. tan θ

2. sin θ

3. cotθ

4. cos θ

Q. Find the area bouded by the curves $y=2x-x^2,4y=(x-2{)}^2$ and $y=0$.

Q. The equation of the normal to the curve $y=|x^2-|x||$ at $x=-2$ is

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

Q. The equation of the circle having one of its diameter as end points of foci of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$, is

1. $x^2+y^2=a^2+b^2$
2. $x^2+y^2=a^2$
3. $x^2+y^2=2a^2$
4. $x^2+y^2=a^2-b^2$

Q.

The equation $z\overline{z}+(2-3i)z+(2+3i)\overline{z}+4$ =0 represents a circle of radius

1. 2

2. 4

3. 3

4. 6

Q. In an ellipse the distance between the foci is 6 and it’s minor axis is 8. Then its eccentricity is

1. 
2. 
3. 
4. 

Q. The equation of the circle passing through the foci 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. $\frac{d}{dx}\left({\cos }^{-1}\frac{1-x^2}{1+x^2}\right)$

1. $2.\frac{1}{1+x^2}$
2. $2.\frac{1}{1-x^2}$
3. $2.\frac{-1}{1+x^2}$
4. $2.\frac{-1}{1-x^2}$

Q. Using the method of integration, find the area bounded by the curve $\left|x\right|+\left|y\right|=1$ .

Q. The area bounded by the curves $y=x(1-lnx)$ and positive x-axis between $x=e^{-1}$ and $x=e$ is

1. $\left(\frac{e^2-4e^{-2}}{5}\right)$
2. $\left(\frac{5e^2-e^{-2}}{4}\right)$
3. $\left(\frac{e^2-5e^{-2}}{4}\right)$
4. $\left(\frac{4e^2-e^{-2}}{5}\right)$

Q. Let $S,S^{\prime }$ be the foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose eccentricity is '$e$'. $P$ is a variable point on the ellipse. Consider the locus of the incentre of the $△PS{S}^{\prime }$

The locus of the incentre of the $△PS{S}^{\prime }$ is

1. ellipse
2. hyperbola
3. parabola
4. circle