Centre of Ellipse

Q.

The point of intersection of the lines represented by the equation $2x^2+3y^2+7xy+8x+14y+8=0$ is

1. $\left(0,2\right)$

2. $\left(1,2\right)$

3. $\left(-2,0\right)$

4. $\left(-2,1\right)$

Q. A normal is drawn to the ellipse $\frac{x^2}{(a^2+4a+5{)}^2}+\frac{y^2}{(a^2+4{)}^2}=1$ whose center is at origin $O$. If the maximum radius of the circle , centered at the origin and touching the normal is $25$ then the positive value of $a$ is

Q. An ellipse has eccentricity $\frac{1}{2}$ and one focus is at the point $P(\frac{1}{2},1)$. If the common tangent to the circle $x^2+y^2=1$ and hyperbola $x^2-y^2=1$ which is nearer to point $P$ is directrix of the given ellipse, then the co-ordinates of centre of ellipse are

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

Q. The locus of the point of intersection of the lines $\left(\sqrt{3}\right)kx+ky-4\sqrt{3}=0$ and $\sqrt{3}x–y–4\left(\sqrt{3}\right)k=0$ is a conic, whose eccentricity is

Q. If the line, $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ meets the plane, $x+2y+3z=15$ at a point P, then the distance of P from the origin is:

1. $2\sqrt{5}$
2. $\frac{7}{2}$
3. $\frac{\sqrt{5}}{2}$
4. $\frac{9}{2}$

Q.

A hyperbola passes through the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with the major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:

1. $\frac{x^2}{9}–\frac{y^2}{4}=1$

2. $\frac{x^2}{9}–\frac{y^2}{16}=1$

3. $x^2–y^2=9$

4. $\frac{x^2}{9}–\frac{y^2}{25}=1$

Q. Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
(i) $x^2+2y^2-2x+12y+10=0$
(ii) $x^2+4y^2-4x+24y+31=0$
(iii) $4x^2+y^2-8x+2y+1=0$
(iv) $3x^2+4y^2-12x-8y+4=0$
(v) $4x^2+16y^2-24x-32y-12=0$
(vi) $x^2+4y^2-2x=0$

Q. If two functions have same Domain and range then the functions are equal functions.

1. False
2. True

Q. The locus of the point of intersection of the straight lines $\frac{x}{a}+\frac{y}{b}=K$ and $\frac{x}{a}-\frac{y}{b}=\frac{1}{K}$ where $K$ is a non-zero real variable, is given by

1. a straight line
2. an ellipse
3. a parabola
4. a hyperbola

Q.

FInd the equation of the tangent to the curve $y=\sqrt{3x-2}$ which is parallel to the line 4x-2y+5=0

Q. Let $y=f\left(x\right)$ be a real-valued differentiable function on the set of all real numbers $R$ such that $f\left(1\right)=1$. If $f\left(x\right)$ satisfies $x{f}^{\prime }\left(x\right)=x^2+f\left(x\right)-2$, then the area enclosed by $y=f\left(x\right)$ with $x$-axis between ordinates $x=0$ and $x=3$ is

Q. The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, centre at the origin and passing through the point (0, 3) is :

1. $xy{y}^{\prime }–y^2+9=0$
2. $xy{y}^{\prime \prime }–x(y^{\prime }{)}^2-y{y}^{\prime }=0$
3. $xy{y}^{\prime }–y^2-9=0$
4. $x+y{y}^{\prime \prime }=0$

Q. STRAIGHT LINES: A right angled triangle ABC having right angle at C, CA = b and CB= a, move such that the angular points A and B slide along the x-axis and y-axis respectively. Find the locus of point C.

Q. A normal is drawn at a point $P(x,y)$ of a curve. It meet the $x-$axis at $Q.$ If $PQ$ is of constant length $k,$ then the differential equation describing such curves is:

1. $\frac{ydy}{dx}=\pm \sqrt{k^2-x^2}$
2. $\frac{ydy}{dx}=\pm \sqrt{k^2-y^2}$
3. $\frac{xdy}{dx}=\pm \sqrt{k^2-x^2}$
4. $\frac{xdy}{dx}=\pm \sqrt{k^2-y^2}$

Q.

The order of the differential equation of the family of all concentric circles centred at $\left(h,k\right)$ is

1. $1$

2. $2$

3. $3$

4. $4$

Q.

If the foci of the ellipse $\frac{x^2}{9}+y^2=1$ subtend right angle at a point $P$. Then, the locus of $P$ is

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

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

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

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

Q.

An ellipse with centre at (0, 0) cuts x axis at (3, 0) and (-3, 0). If its eccentricity is $\frac{1}{2}$ then the length of its semiminor axis is

1. 

2. 3√3/2

3. 

4. 

Q.

Which of the following statements are true and which are false? In each case give a valid reason for saying so

$r:$ Circle is a particular case of an ellipse.

1. True
2. False

Q.

Form the differential equation of the family of ellipse having foci on Y-axis and centre at origin.

Q. Which of the following is correct for a real function?

1. Range of a real function should be subset of $R$
2. If $f:R\to R$ and $f\left(x\right)=x^3-2|x|$, then it is real function
3. If $f:R\to R$ and $f\left(x\right)=1-x^2$, then it is real function
4. Domain of a real function should be subset of $R$

Q.

Form the differential equation of the family of hyperbolas having foci on X-axis and centre at origin.

Q. $f\left(x\right)={\tan }^{2}x.{\sin }^{2}x$ and $g\left(x\right)={\tan }^{2}x-{\sin }^{2}x$ are equal functions.

1. False
2. True

Q. A real function will always be a real valued function but the converse is not necessarily true.

1. True
2. False

Q. A function which has its domain and ranges as either $R$ or any subset of $R$ is called a real function.

1. False
2. True

Q. Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+m-n=0$ and $l^2+m^2-n^2=0$. Then the value of ${\sin }^4\alpha +{\cos }^4\alpha$ is :

1. $\frac{3}{4}$
2. $\frac{1}{2}$
3. $\frac{5}{8}$
4. $\frac{3}{8}$

Q. Find the centre, the lengths of the axes, eccentricity, foci of the ellipse:
$\left(v\right)4x^2+16y^2-24x-32y-12=0$

Q. Find the equation of the ellipse whose vertices are $(\pm 6,0)$ and foci are $(\pm 4,0)$

Q. Find the equation of the ellipse whose foci are $(0,\pm 3)$ and length of the minor axis is $16$

Q.

Find the equation of a curve passing through the point (0, 0) and whose differential equation is $y^{\prime }=e^{x}sinx$

Q. The equation of the ellipse having vertices at $(\pm 5,0)$ and foci $(\pm 4,0)$ is

1. $9x^2+25y^2=225$
2. $\frac{x^2}{25}+\frac{y^2}{16}=1$
3. $4x^2+5y^2=20$
4. $\frac{x^2}{9}+\frac{y^2}{25}=1$