Standard Equation of Parabola

Q.

If one end of a focal chord $AB$ of the parabola $y^2=8x$ is at $A\left(\frac{1}{2},-2\right)$, then the equation of a tangent to it at $B$ is

1. $x+2y+8=0$

2. $2x-y-24=0$

3. $x-2y+8=0$

4. $2x+y-24=0$

Q. $M$ is the foot of perpendicular from a point $P$ on a parabola $y^2=4ax$ to its directrix and $SPM$ is an equilateral triangle, where $S$ is the focus of the parabola. Then the value of $\frac{SP}{a}$ is

Q.

Find the equation of a parabola with vertex at the origin, the axis along x-axis and passing through (2, 3).

Q.

The focus and directrix of a parabola are (1, -1) and x+y+3=0. Its vertex is

1. 

2. 

3. 

4. 

Q.

The parabola $y^2=x$ is symmetric about

1. Both x – axis and y – axis

2. The line y = x

3. x – axis

4. y – axis

Q. The area of the triangle formed by three points on a parabola is _____ the area of the triangle formed by the tangents at these three points.

1. Equal

2. twice

3. thrice

4. four times

Q. The points on the parabola $y^2=36x$ whose ordinate is three times the abscissa are

1. None of these
2. (0, 0) (4, 12)
3. (4, 12)
4. (1, 3), (4, 12)

Q.

If V and S are respectively the vertex and focus of the parabola $y^2+6y+2x+5=0,$ then SV=

1. 2

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

3. none of these

4. 1

Q.

Find the equation of a parabola with vertex at the origin and the directrix, y=2.

Q. The coordinates of the end points of the latus rectum of parabola $x^2+8y=0$ are $\underset{–}{}$

Q. If point $(4,4)$ lies on a parabola whose focus lies on $x$ -axis and directrix is $lx+my=1$ such that $(l,m)$ lies on curve $x^2+y^2-32xy+8x+8y-1=0,$ then

1. Co-ordinates of focus may be $(5,0)$
2. Co-ordinates of focus may be $(3,0)$
3. If tangent at $(4,4)$ passes through origin, value of $m$ may be $\frac{1}{3}$
4. If tangent at $(4,4)$ passes through origin, equation of axis may be $x-4y-3=0$

Q. The vertex of the conic represented by $25(x^2+y^2)=(3x-4y+12{)}^2$ is

1. $(\frac{24}{25},-\frac{18}{25})$
2. $(-\frac{36}{25},\frac{48}{25})$
3. $(-\frac{18}{25},\frac{24}{25})$
4. $(0,0)$

Q. If $y_1,y_2$ and $y_3$ are the ordinates of the vertices of a triangle inscribed in the parabola $y^2=4ax$ then its area is

1. 
2. 
3. none
4. 

Q. Write the axis of symmetry of the parabola $y^2=x$.

Q. The general solution of the differential equation $\frac{dy}{dx}+\sin \frac{x+y}{2}=\sin \frac{x-y}{2}$ is (where $c$ is a constant of integration)

1. $\ln \left|\tan \left(\frac{y}{2}\right)\right|=c-2\sin x$
2. $\ln \left|\tan \left(\frac{y}{4}\right)\right|=c-2\sin \left(\frac{x}{2}\right)$
3. $\ln \left|\tan (\frac{y}{2}+\frac{\pi }{4})\right|=c-2\sin x$
4. $\ln \left|\tan (\frac{y}{2}+\frac{\pi }{4})\right|=c-2\sin \left(\frac{x}{2}\right)$

Q. Find the equation of the parabola whose-
Vertex $(0,0)$; Focus $(3,0)$

Q. Consider the two curves
$C_1:y^2=4x,C_2:x^2+y^2-6x+1=0.$
Then,

1. $C_1$ and $C_2$ touch each other only at one point
2. $C_1$ and $C_2$ touch each other exactly at two points
3. $C_1$ and $C_2$ neither intersect nor touch each other
4. $C_1$ and $C_2$ intersect (but do not touch) at exactly two points

Q.

Solution of the equation $cosy=cosx$

1. $y=n\pi +(-1{)}^{n}x$

2. $n\pi +x$

3. $n\pi +x$

4. $y=2n\pi +x$

Q. In the parabola, $y^2-2y+8x-23=0$, the length of double ordinate at a distance of 4 units from its vertex is

1. 
2. 
3. 
4. 

Q. If one vertex of a chord of parabola $y^2=8ax$ is at $(0,0),$ then the locus of a point which divides the chord in the ratio $1:2$ is

1. $y^2=4ax$
2. $y^2=8ax$
3. $y^2=\frac{8a}{3}x$
4. $y^2=2ax$

Q. The area of the triangle formed by the intersection of a line parallel to $x$-axis and passing through $P(h,k)$ with the lines $y=x$ and $x+y=2$ is $4h^2$. Find the locus of the point $P$.

Q. Find the equation of the parabola with
(i) vertex $(0,0)$ and focus $(3,0)$

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

Q. The equation of the parabola whose vertex and focus lies on the x – axis at distance a and a’ from the origin, is

1. 
2. 
   
3. 
4. 

Q. The solution of the differential equation $\frac{dy}{dx}=\frac{1}{xy[x^2\sin y^2+1]}$ is
(where $C$ is integration constant)

1. $x^2(\cos y^2-\sin y^2-C{e}^{-y^2})=2$
2. $y^2(\cos x^2-\sin y^2-C{e}^{-y^2})=2$
3. $x^2(\cos y^2-\sin y^2-e^{-y^2})=4C$
4. $x(\cos y^2-\sin y^2-e^{-y})=4C$

Q. If $y_1,y_2$ and $y_3$ are the ordinates of the vertices of a triangle inscribed in the parabola $y^2=4ax$ then its area is

1. $\frac{1}{2a}|(y_1-y_2)(y_2-y_3)(y_3-y_1)|$
2. $\frac{1}{4a}|(y_1-y_2)(y_2-y_3)(y_3-y_1)|$
3. $\frac{1}{8a}|(y_1-y_2)(y_2-y_3)(y_3-y_1)|$
4. none

Q. Find the equation of a parabola
(i) having its vertex ar A (1, 0) and focus at S(3, 0)
(ii) having its focus at S(2, 0) and one extremity of its latus rectum at (2, 2)
(iii) having focus at (0, -3) and directrix y=3

Q.

Which of the following statements are correct?

1. The focus of $x^2=4ay$ is $(0,a)$

2. The directrix of $y^2=-4ax$ is $x+a=0$

3. The end points of latus rectum of $x^2=-4ay$ is (a, 2a) and (a, -2a)

4. The parabolas $x^2=4ay\text{and}{y}^2=4ax$ are equal.

Q.

There is a parabola having axis as $x$-axis, vertex is at a distance of $2$ unit from origin & focus is at $\left(4,0\right)$.

Which of the following point does not lie on the parabola.

1. $\left(6,8\right)$

2. $\left(5,2\sqrt{6}\right)$

3. $\left(8,4\sqrt{3}\right)$

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

Q.

The equation of the parabola with its vertex at the origin, axis on the y-axis and passing through the point (6, -3) is

1. 

2. 

3. 

4.