Parametric Equation of Parabola

Q.

Calculate the ratio in which the line joining $A(-4,2)$ and $B(3,6)$ is divided by a point $P(x,3)$. Also find
(i) $x$.
(ii) length of $AP$.

Q. If $Q$ be a point on the parabola $y^2=8x$ and $P(-2,0)$ be a point in the $xy$ plane. If the locus of the mid point of $PQ$ is a parabola, then its focus is

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

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.

(I) Write down the coordinates of the point P, that divides the line joining $A(-4,1)$and $B(17,10)$ in the ratio $1:2$.

(II) Calculate the distance OP, where O is origin.

(III) Find the equation of a line AB.

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. Let P be the point on the parabola, $y^2=8x$, which is at a minimum distance from the centre C of the circle, $x^2+(y+6{)}^2=1$. Then, the equation of the circle, passing through C and having its centre at P is

1. $x^2+y^2-4x+8y+12=0$
2. $x^2+y^2-x+4y-12=0$
3. $x^2+y^2-\frac{x}{4}+2y-24=0$
4. $x^2+y^2-4x+9y+18=0$

Q. The curve described parametrically by $y=2at$ and $x=a{t}^2$ represents

1. an ellipse
2. a parabola
3. a hyperbola
4. a circle

Q. If $\theta$ be the angle subtended at the focus by the chord which is normal at the point $(\lambda ,\lambda ),\lambda \ne 0$ to the parabola $y^2=4x,$ then the equation of the line making angle $\theta$ with positive $x-$axis and passing through $(1,2)$ is

1. $y=2$
2. $x+2y=5$
3. $x+y=3$
4. $x=1$

Q. If the tangent at $P$ on $y^2=4ax$ meets the tangent at the vertex in $Q$, and $S$ is the focus of the parabola, then $\angle SQP=$

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

Q. A focal chord of the parabola $y^2=4ax$ meets it at P and Q. If S is the focus then $\frac{1}{SP}+\frac{1}{SQ}=$

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

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 locus of a point on the variable parabola $y^2=4ax$, whose distance from focus is always equal to $k$, is equal to

1. $4x^2+y^2-4kx=0$
2. $x^2+y^2-4kx=0$
3. $2x^2+4y^2-8kx=0$
4. $4x^2-y^2+4kx=0$

Q. The parametric equation of a parabola is $x=t^2+1,y=2t+1$ Then which of the following options are correct ?

1. Vertex is $(1,1)$
2. Vertex is $(-1,-1)$
3. Length of latus rectum of parabola is $8$ units
4. Length of latus rectum of parabola is $4$ units

Q.

The points of intersection of the curves whose parametric equations are $x=t^2+1,y=2tandx=2s,y=\frac{2}{s}$ Is given by

1. (1, - 3)

2. (2, 2)

3. (-2, 4)

4. (1, 2)

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Tangents are drawn from the point}(2,3)& \text{(P)}& (9,-6)\\ & \text{to the parabola}{y}^2=4x.\text{Then point(s) of}& & \\ & \text{contact is (are)}& & \\ \text{(B)}& \text{From a point}P\text{on the circle}{x}^2+y^2=5,& \text{(Q)}& (1,2)\\ & \text{the equation of chord of contact to the}& & \\ & \text{parabola}{y}^2=4x\text{is}y=2(x-2).\text{Then}& & \\ & \text{the coordinates of}P\text{are}& & \\ \text{(C)}& P(4,-4),Q\text{are points on parabola}& \text{(R)}& (-2,1)\\ & y^2=4x\text{such that area of}△POQ\text{is}6& & \\ & \text{sq. units where}O\text{is the vertex. Then}& & \\ & \text{coordinates of}Q\text{may be}& & \\ \text{(D)}& \text{The common chord of circle}{x}^2+y^2=5& \text{(S)}& (4,4)\\ & \text{and parabola}6y=5x^2+7x\text{will pass}& & \\ & \text{through point(s)}& & \\ & & \text{(T)}& (-2,2)\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(Q)},\text{(S)}$
2. $\text{(C)}\to \text{(P)},\text{(T)}$
3. $\text{(D)}\to \text{(P)},\text{(S)}$
4. $\text{(D)}\to \text{(Q)},\text{(R)}$

Q. The parametric equation of the curve $y^2=8x$ are

1. $x=t^2,y=2t$
2. $x=2t^2,y=4t$
3. $x=2t,y=4t^2$
4. None of these

Q. If the tangent at $P$ on $y^2=4ax$ meets the tangent at the vertex in $Q$, and $S$ is the focus of the parabola, then $\angle SQP=$

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

Q. The parametric point $(3+t^2,3t-2)$ represents a parabola with

1. focus at $(-3,-2)$
2. vertex at $(3,-2)$
3. directrix $x=-5$
4. all of these

Q.

Which of the following can be the parametric equation of
$(x-1{)}^2=-36(y-4)$

1. $x=1+18t$ and$y=4-9t^2$

2. $x=-1+18t$ and$y=-4-9t^2$

3. $x=-1-18t$ and $y=4-9t^2$

4. $x=1+18t$ and $y=-4-9t^2$

Q. The length of the chord of the parabola $x^2=4y$ passing through the vertex and having slope $cot\alpha$ is

1. $4cos\alpha .cose{c}^2\alpha$
2. $4tan\alpha sec\alpha$
3. $4sin\alpha .se{c}^2\alpha$
4. $noneofthese$

Q. Find the parametric equation of the parabola $x^2-4x-32=-12y$

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

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 parabola
4. a hyperbola

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. 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. 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. Two distinct chords of the parabola $y^2=4ax$ passing through $P(a,2a)$ are bisected by the line $x-y+1=0$. The possible length of the latus rectum of the parabola when $a>0$ is

1. $2$
2. $3$
3. $4$
4. $5$

Q. The triangle PQR of area A is inscribed in the parabola $y^2=4ax$ such that P lies at the vertex of the parabola and base QR is a focal chord. The numerical difference of the ordinates of the points Q & R is

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

Q. The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $x^2=4y$, internally in the ratio $1:2$, is:

1. $9x^2-12y=8$
2. $4x^2-3y=2$
3. $x^2-3y=2$
4. $9x^2-3y=2$

Q. If $\theta$ be the angle subtended at the focus by the chord which is normal at the point $(\lambda ,\lambda ),\lambda \ne 0$ to the parabola $y^2=4x,$ then the equation of the line making angle $\theta$ with positive $x-$axis and passing through $(1,2)$ is

1. $y=2$
2. $x+2y=5$
3. $x+y=3$
4. $x=1$

Q. The parametric equation of a parabola is $x=t^2+1,y=2t+1$ Then which of the following options are correct ?

1. Vertex is $(1,1)$
2. Vertex is $(-1,-1)$
3. Length of latus rectum of parabola is $8$ units
4. Length of latus rectum of parabola is $4$ units