Four Common Forms of Parabola Equation

Q. The focus of parabola $x^2=-16y$ is​

1. (4, 0)
2. (0, 4)
3. (-4, 0)
4. (0, -4)

Q. The equation of parabola whose vertex and focus are (0, 4) and (0, 2) respectively, is

1. 
   
2. 
   
3. 
4. 

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. 

Q. Vertex of the parabola $9x^2-6x+36y+9=0$ is

1. 
   
2. 
3. 
4. 
   

Q. Vertex of the parabola $y^2+2y+x=0$ lies in the quadrant

1. Fourth
2. First
3. Third
4. Second

Q. Focus and directrix of the parabola $x^2=-8ay$ are

1. (0, 2a) and y = - 2a
2. (0, -2a) and y = 2a
3. (-2a, 0) and x = 2a
4. (2a, 0)and x = - 2a

Q.

The vertex of a parabola is the point (a, b) and latus rectum is of length /. If the axis of the parabola is along the positive direction of y-axis, then its equation is

1. 

2. 

3. 

4. 

Q. If an equilateral triangle is inscribed in a parabola $y^2=12x$ with one vertex at the vertex of the parabola, then its height is

1. $24\sqrt{3}$
2. $16\sqrt{3}$
3. $24$
4. $36$

Q. The length of the latus rectum of the parabola $25\left[\right(x-2{)}^2+(y-4{)}^2]=(4x-3y+12{)}^2$ is

1. $\frac{16}{5}$
2. $\frac{12}{5}$
3. $\frac{8}{5}$
4. $\frac{4}{5}$

Q. The end points of latus rectum of the parabola $x^2=4ay$ are

1. (a, 2a), (2a, - a)
2. (-a, 2a), (2a, a)
3. (a, -2a), (2a, a)
4. (-2a, a) , (2a, a)

Q. An equilateral triangle is inscribed in the parabola $y^2=4ax$ such that one vertex of this triangle coincides with the vertex of the parabola. The length of side of this triangle is

1. $2a\sqrt{3}$
2. $4a\sqrt{3}$
3. $8a\sqrt{3}$
4. $6a\sqrt{3}$

Q.

Let $y=mx+c,m>0$ be the focal chord of $y^2=-64x$ which is tangent to ${(x+10)}^2+y^2=4$. Then the value of $4\sqrt{2}(m+c)$ is equal to

Q. The ends of latus rectum of parabola $x^2+8y=0$

1. (-4, -2) and (-4, 2)
2. (-4, -2) and (4, -2)
3. (4, -2) and (-4, 2)
4. (-4, -2) and (4, 2)

Q. For the given parabola $9y^2-16x-12y-57=0$, which of the following is/are correct?

1. axis will be $y=2$
2. axis will be $3y=2$
3. vertex will be $(61,2)$
4. vertex will be $(-\frac{61}{16},\frac{2}{3})$

Q. The equation of the parabola whose vertex is (-1, -2) axis is vertical and which passes through the point (3, 6) is

1. 
2. 
3. 
4. None of these

Q. Area bounded by the curve $f\left(x\right)=\frac{1}{x^2+[x{]}^2+2\{x\}+1-2x[x]}$ and x-axis between$x=\frac{-3}{2}$ and $x=\frac{5}{2}$ is equal to
(where [ ] denotes greatest integer function and { } denotes fractional part function)

Q. The equation of the latus rectum of the parabola $x^2+4x+2y=0$ is

1. 3y = 2
2. 2y = 3
3. 3y + 2 = 0
4. 2y + 3 = 0

Q. The equation of the parabola whose focus lies at the intersection point of the lines $x+y=3$ and $x-y=1$ and directrix is $x-y+5=0$

1. $x^2+y^2+2xy+18x+6y-25=0$
2. $x^2+y^2+2xy+18x+6y+25=0$
3. $x^2+y^2+2xy-18x+6y-15=0$
4. $x^2+y^2+2xy-18x-6y+15=0$

Q. A focal chord of the parabola $y^2=8x$ is tangent to the circle $(x-5{)}^2+y^2=3$, then the possible value of the twice of square of slope of this chord is

Q. Equation of the parabola whose vertex is $(-3,-2)$, axis is horizontal and which passes through the point $(1,2)$ is

1. $y^2+4y+4x-8=0$
2. $y^2+4y-4x+8=0$
3. $y^2+4y-4x-8=0$
4. none of these

Q. The focus of parabola $x^2=-16y$ is​

1. (4, 0)
2. (0, 4)
3. (0, -4)
4. (-4, 0)

Q. The equations $x=\frac{t}{4},y=\frac{t^2}{4}$ represents

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

Q. If the radius of a circle whose centre is $(x_0,y_0)$ touches a parabola $y^2=4x$, a straight line $x–y+1=0$ and the $y-$axis is $\sqrt{p}\tan \left(\frac{\pi }{8}\right)$ where $x_0–y_0+1<0,$ then the value of $p$ is

Q. The end points of latus rectum of the parabola $x^2=4ay$ are

1. (a, 2a), (2a, - a)
2. (-a, 2a), (2a, a)
3. (a, -2a), (2a, a)
4. (-2a, a) (2a, a)

Q. The equation of the parabola whose axis is parallel to y – axis and passing through (4, 5), (–2, 11), (–4, 21) is

1. $x^2-4x-2y+10=0$
2. $x^2-2x-y+5=0$
3. $x^2-4x-2y+10=0$
4. $y^2-2x3y+4=0$

Q. Axis of the parabola $x^2-4x-3y+10=0$ is

1. x – 2 = 0
2. y + 2 – 0
3. x + 2 – 0
4. y – 2 = 0

Q. Let a circle $x^2+y^2-2x-3=0$ touches the directrix of a parabola and passes through end points of latus rectum of the same parabola. If latus rectum of the parabola is chord of maximum length with respect to given circle and equation of parabola is $y^2=kx$, then $k=$

Q. The equation of the parabola whose axis is parallel to y – axis and passing through (4, 5), (–2, 11), (–4, 21) is

1. $x^2-4x-2y+10=0$
2. $x^2-2x-y+5=0$
3. $x^2-4x-2y+10=0$
4. $y^2-2x3y+4=0$

Q. The equation of parabola whose vertex and focus are (0, 4) and (0, 2) respectively, is

1. $y^2-8x=32$
   
2. $y^2+8x=32$
   
3. $x^2+8y=32$
4. $x^2-8y=32$

Q. The length of the latus rectum of the parabola $25\left[\right(x-2{)}^2+(y-4{)}^2]=(4x-3y+12{)}^2$ is

1. $\frac{16}{5}$
2. $\frac{12}{5}$
3. $\frac{8}{5}$
4. $\frac{4}{5}$