Parametric Equation of Parabola

Q.

The circle circumscribing the triangle formed by three tangents to a parabola passes through _____

1. Focus of the parabola

2. Vertex of the parabola

3. Extremeties of latus rectum

4. Origin

Q. $x-2=t^2,y=2t$ are the parametric equations of the parabola

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

Q. The curve described parametrically by $x=t^2+t+1,y=t^2-t+1$ represents

1. a pair of straight lines
2. an ellipse
3. a hyperbola
4. a parabola

Q. Consider a parabola $y^2=4x$, Let $A$ be the vertex of parabola, $P$ be any point on the parabola and $B$ is a point on the axis of parabola, if $PA\perp PB$, then the locus of centroid of $△PAB$ is

1. $9y^2=-6x-8$
2. $9y^2=6x+8$
3. $9y^2=6x-8$
4. $9y^2=-6x+8$

Q. Vertex of the parabola whose parametric equation is $x=t^2-t+1,y=t^2+t+1;t\in R,$ is

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

Q. If $t_1$ and $t_2$ are the extremities of any focal chord of the parabola $y^2=4ax$ then $t_1{t}_2=$

1. $\pm 1$
2. $0$
3. $\frac{1}{2}$
4. $-1$

Q. Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola $y^2=4x$. If $X$ is any point on the arc $POQ$ of the parabola, where $O$ is the vertex such that the area of $△PXQ$ is maximum, then $4$ times the maximum area (in sq. units) is

Q. If one end point of the focal chord of the parabola $y^2=4ax$ is $(1,2)$, then second end point lies on

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

Q. If the ends of a focal chord of the parabola $y^2=4ax$ are $(x_1,y_1)$ and $(x_2,y_2)$ then $x_1,x_2+y_1{y}_2=$

1. $a^2$
2. $-3a^2$
3. $5a^2$
4. $-5a^2$

Q. A normal chord $AB$ of a parabola $y^2-12x=0$ subtends a right angle at the vertex of the parabola. If the point of intersection of the normals drawn at $A$ and $B$ is $(p,q)$, then the value of $\frac{p^2}{q^2}$ is

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

Q. Find equation of the line which is equidistant from parallel lines $9x+6y-7$ $=$ $0$ and $3x+2y+6$ $=$ $0$.

Q. An equilatral triangle inscribed in parabola $y^2=4ax$ whose one vertex is at the vertex of parabola. Then the length of the side of the triangle is

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

Q. Coordinates of parametric point on the parabola, whose focus is $(\frac{-3}{2},-3)$ and the directrix is $2x+5=0$ is given by

1. $(2t^2+2,2t-3)$
2. $(\frac{1}{2}{t}^2-2,t-3)$
3. $(\frac{1}{2}{t}^2-2,t+3)$
4. $(\frac{1}{2}{t}^2+2,t+3)$

Q.

Match the following equation of parabola

$\begin{array}{cc}\text{Parabola}& \text{parametric equation}\\ \text{N)}{x}^2=4ay& \text{1)}(a{t}^2,2at)\\ \text{E)}{y}^2=4ax& \text{2)}(-at,2at)\\ \text{W)}{y}^2=-4ax& \text{3)}(2at,a{t}^2)\\ \text{S)}{x}^2=-4ay& \text{4)}(2at,a{t}^2)\end{array}$

1. N-2, E-3, W-1, S - 4

2. N-1, E-3, W-2, S - 4

3. N-4, E-1, W-2, S - 3

4. N-3, E-1, W-2, S - 4

Q.

$x^2+y^2\mp 2kx\mp 2ky+k^2$ is a set of circles. Which of the following statements are true about them?

1. All of them touches x-axis

2. All of them touches y-axis

3. They are concentric

4. The radius of all the circles are same

Q. If the lengths of the chords intercepted by the circle $x^2+y^2+2gx+2fy=0$ from the coordinate axes are $10$ and $24$ units, respectively, then the radius of the circle is:

1. $17$
2. $9$
3. $14$
4. $13$

Q. If $x=t^2+2$ and $y=2t$ represent the parametric equation of the parabola

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

Q. If $2p$ is the length of perpendicular from the origin to the line whose intercepts on the axes are $2a$ and $b$, then show that $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$.

Q. The Cartesian equation of the curve whose parametric equations are $x=t^2+2t+3$ and y = t + 1 is

1. $y=(x-1{)}^2+2(y-1)+3$
2. $x=(y-1{)}^2+2(y-1)+5$
3. $x=y^2+2$
4. $\text{none of these}$

Q. The parametric equation of parabola $(y-2{)}^2=12(x-4)$ is

1. $x=4-3t^2,y=2-6t$
2. $x=2+3t,y=4+t^2$
3. $x=4+3t^2,y=2+6t$
4. $x=2+3t^2,y=4+6t$

Q. The curve described parametrically by
$x=t^2+t+1,y=t^2-t+1$ represents

1. a pair of straight lines
2. an ellipse
3. a parabola
4. a hyperbola