General Equation of Parabola

Q. The coordinates of the point on the parabola $y^2=18x$ whose ordinate is three times the abscissa, are $\underset{–}{}$

Q.

What are the parts of a hyperbola?

Q. The equation of the parabola whose axis is vertical and passes through the points (0, 0), (3, 0) and (-1, 4) is

1. 
2. 
   
3. 
   
4. 

Q.

The equation of the parabola whose focus is (1, -1) and the directrix is x+y+7=0 is

1. $x^2+y^2-2xy-18x-10y=0$

2. $x^2-18x-10y-45=0$

3. $x^2+y^2-18x-10y-45=0$

4. $x^2+y^2-2xy-18x-10y-45=0$

Q.

The equation $16x^2+y^2+8xy-74x-78y+212=0$ represents

1. a circle

2. a parabola

3. an ellipse

4. a hyperbola

Q. $y=f\left(x\right)$ is the parabola of the form $y=x^2+ax+1,$ its tangent at the point of intersection of $y-$axis and parabola also touches the circle $x^2+y^2=r^2.$ It is known that no point of the parabola is below $x-$axis. The radius of the circle (in units) when $a$ attains its maximum value is

1. $\frac{1}{\sqrt{10}}$
2. $1$
3. $\frac{1}{\sqrt{5}}$
4. $\sqrt{5}$

Q.

Find the equation of the parabola whose focus is (5, 2) and having vertex at (3, 2).

Q.

Find the equation of the parabola whose:

(i) focus is (3, 0) and the directrix is 3x+4y=1

(ii) focus is (1, 1) and the directrix is x+y+1=0

(iii) focus is (0, 0) and the directrix is 2x-y-1=0

(iv) focus is (2, 3) and the directrix is x-4y+3=0

Q. The locus of the midpoint of the focal distance of a variable point moving on the parabola $y^2=4ax$ is a parabola whose

1. Latus rectum is half the latus rectum of the original parabola
2. Vertex is $(\frac{a}{2},0)$
3. Directrix is $y-$axis
4. Focus has the co-ordinates $(a,0)$

Q. Consider a circle with its centre lying on the focus of the parabola $y^2=2px$ such that it touches the directrix of the parabola. Then the point of intersection of the circle and parabola can be

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

Q. The equation of the parabola whose vertex is $(-3,0)$ and directrix is $x+5=0,$ is

1. $x^2=8(y+3)$
2. $y^2=8(x+3)$
3. $x^2=8(y-3)$
4. $y^2=8(x-3)$

Q.

If the points (0, 4) and (0, 2) are respectively the vertex and focus of the parabola, then find the equation of the parabola.

Q.

Find the equation of the parabola whose focus is the point (2, 3) and directrix is the line x-4y+3=0.Also, find the length of its latus-rectum.

Q.

The equation of the parabola whose vertex is (a, 0) and the directrix has the equation x+y=3a, is

1. none of these

2. $x^2+y^2+2xy+6ax+10ay+7a^2=0$

3. $x^2-2xy+y^2+6ax+10ay-7a^2=0$

4. $x^2-2xy+y^2-6ax+10ay-7a^2=0$

Q.

Write the equation of the parabola with focus (0, 0) and directrix x+y-4=0.

Q.

If $P=(1,0),Q=(-1,0)$ and $R=(2,0)$ are three given points, then the locus of the point $S$ satisfying the relation ${\left(SQ\right)}^2+{\left(SR\right)}^2=2{\left(SP\right)}^2$, is

1. A straight line parallel to x-axis

2. A circle passing through the origin

3. A circle with the center at the origin

4. A straight line parallel to the y-axis

Q.

What is the focus of a parabola?

Q.

If the line $y=mx+a$ meets the parabola $y^2=4ax$ at two points whose abscissas are $x_1$ and $x_2,$ then $x_1+x_2=0$ if

1. $m=1$
2. $m=2$
3. $m=-\frac{1}{2}$
4. $m=-1$

Q. Equation of the parabola, if coordinates of vertex and focus are $(0,0)$ and $(2,3)$ respectively, is

1. $4x^2+9y^2-156x-104y-12xy=0$
2. $9x^2+4y^2-104x-156y-12xy=0$
3. $9x^2+4y^2-156x-104y-12xy=0$
4. $9x^2+4y^2-104x+156y+12xy=0$

Q.

If $t_1$and $t_2$ are the parameters of the endpoints of a focal chord for the parabola $y^2=4ax$, then which one is correct.

1. $t_1{t}_2=1$

2. $\frac{t_1}{t_2}=1$

3. $t_1{t}_2=-1$

4. $t_1+t_2=1$

Q. If equation of the parabola is $(6x+8y-5{)}^2=60(8x-6y+9),$ then

1. Equation of axis of the parabola is $6x+8y-5=0$
2. Equation of tangent at vertex of the parabola is $6x+8y-5=0$
3. Length of latus rectum is $\frac{3}{2}$ units
4. Length of latus rectum is $6$ units

Q. If coordinates of vertex and focus of a parabola are $(1,2)$ and $(4,6)$ respectively, then

1. equation of the parabola is $(4x-3y+2{)}^2=100(3x+4y-11)$
2. equation of the parabola is $(3x+4y-11{)}^2=100(4x-3y+2)$
3. length of latus rectum of the parabola is $5$ units
4. length of latus rectum of the parabola is $20$ units

Q. Line $y=2x-p$ cuts the parabola $y=x^2-4x$ at points $A$ and $B$. If $\angle AOB=\frac{\pi }{2}$, where $O$ is the origin, then the value of $p$ is

Q.

The equation of the parabola with focus (0, 0) and directrix x+y=4 is

1. $x^2+y^2-2xy+8x+8y-16=0$

2. $x^2+y^2-2xy+8x+8y=0$

3. $x^2+y^2+8x+8y-16=0$

4. $x^2-y^2+8x+8y-16=0$

Q.

Write the equation of the parabola whose vertex is at (-3, 0) and the directrix is x+5=0

Q. Let $P$ be the foot of the perpendicular from focus $S$ of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ on the line $bx-ay=0$ and let $C$ be the centre of the hyperbola. Then the area of the rectangle whose sides are equal to that of $SP$ and $CP$ is

1. $2ab$
2. $ab$
3. $\frac{a^2+b^2}{2}$
4. $\frac{a}{b}$

Q. The axis of the parabola $x^2-4x-y+1=0$ is

1. y=-3
2. x=-3
3. x=2
4. none of these

Q. The equation of the parabola whose axis is vertical and passes through the points (0, 0), (3, 0) and (-1, 4) is

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

Q. Equation of the parabola with its vertex at $(1,1)$ and focus $(3,1)$ is

1. $(x1{)}^2=8(y-1)$
2. $(y-1{)}^2=8(x-3)$
3. $(y-1{)}^2=8(x-1)$
4. $(x-3{)}^2=8(y-1)$

Q. Tangents and normal drawn to parabola $y^2=4ax$ at point $P(a{t}^2,2at),t\ne 0$, meet the x-axis at $T$ and $N$, respectively. If $S$ is the focus of the parabola, then

1. $SP\ne ST=SN$
2. $SP=ST\ne SN$
3. $SP\ne ST\ne SN$
4. $SP=ST=SN$