The normal at a point on the parabola $y^2=4 x$ passes through a point $P$. Two more normals to this parabola also pass through $P$. If the centroid of the triangle formed by the feet of these three normals is $G(2,0)$, then the abscissa of $P$ is

A normal chord $P Q$ drawn at a point $P$ on the parabola $y^2=5 x$ subtends a right angle at the vertex. If $P$ lies in the first quadrant, then the other end $Q$ of the normal chord is

If $L(p, q), q>3$ is one end of the latus rectum of the parabola $(y-2)^2=3(x-1)$, then the equation of the tangent at $L$ to this parabola is

The number of normals that can be drawn through the point $(2,0)$ to the parabola $y^2=7 x$ is

If $m_1$ and $m_2$ are the slopes of the tangents drawn from the point $(1,4)$ to the parabola $y^2=11 x$, then $2\left(m_1^2+m_2^2\right)=$

If the normals drawn at the points $P\left(\frac{3}{4}, \frac{3}{2}\right)$ and $Q(3,3)$ on the parabola $y^2=3 x$ intersect again on $y^2=3 x$ at $R$, then $R=$

If $\theta$ is the acute angle between the tangents drawn from the point $(1,5)$ to the parabola $y^2=9 x$, then

For the parabola $y=x^2-3 x+2$, match the items in List I to that of the items in List II. $S$ is a focus, $Z$ is intersection of axis and directrix, $P$ is one end of latus rectum, $Q$ is the point on the parabola at which tangent is parallel to $X$-axis.

$ \begin{array}{llll} \hline & \text { List I } & & \text { List II } \\ \hline \text { A. } & P & \text { I. } & (2,0) \\ \hline \text { B. } & Q & \text { II. } & \left(\frac{3}{2},-\frac{1}{4}\right) \\ \hline \text { C. } & S & \text { III. } & \left(\frac{3}{2}, 0\right) \\ \hline \text { D. } & Z & \text { IV. } & \left(\frac{3}{2},-\frac{1}{2}\right) \\ \hline & & \text { V. } & \left(0, \frac{3}{2}\right) \\ \hline \end{array} $

The locus of a point which divides the line segment joining the focus and any point on the parabola $y^2=12 x$ in the ratio $m: n(m+n \neq 0)$ is a parabola.

Then, the length of the latus rectum of that parabola is

If the normal drawn at $P(8,16)$ to the parabola $y^2=32 x$ meets the parabola again at $Q$, then the equation of the tangent drawn at $Q$ to the parabola is

The focal distance of a point $(5,5)$ on the parabola $x^2-2 x-4 y+5=0$ is

Consider the parabola $25\left[(x-2)^2+(y+5)^2\right]=(3 x+4 y-1)^2$, match the characteristic of this parabola given in List I with its corresponding item in List II.

$ \begin{array}{lll} \hline & \text { List I } & \text { List II } \\\\ \hline \text { I } & \text { Vertex } & \text { (A) } 8 \\\\ \hline \text { II } & \text { length of latus rectum } & \text { (B) }\left(\frac{29}{10}, \frac{-38}{10}\right) \\\\ \hline \text { III } & \text { Directrix } & \text { (C) } 3 x+4 y-1=0 \\\\ \hline \text { IV } & \begin{array}{l} \text { One end of the latus } \\\\ \text { rectum } \end{array} & \text { (D) }\left(\frac{-2}{5}, \frac{-16}{5}\right) \\\\ \hline \end{array} $

The correct answer is

If $\mathbf{A B}$ is the focal chord of the parabola $y^2=16 x$ and $A=(1,-4)$, then the equation of the normal to the parabola at the point $B$ is

If one of the vertices of an equilateral triangle inscribed in the parabola $y^2=12 x$ coincides with the vertex of the parabola, then the area (in sq units) of that triangle is

If $x-2 y+k=0$ is a tangent to the parabola $y^2-4 x-4 y+8=0$, then the value of $k$ is

If the points of intersection of the parabolas $y^2=5 x$ and $x^2=5 y$ lie on the line $L$, then the area of the triangle formed by the directrix of one parabola, latus rectum of another parabola and the line $L$ is

If the line $2 x+3 y+n=0$ is a tangent to the parabola $y^2=8 x$, then the equation of the normal drawn at the point $(2 n, 4 \sqrt{n})$ to the parabola $y^2=8 x$ is

$a x-y+c=0$ is the equation of the common tangent to the parabola $y^2=8 \sqrt{5} x$ and the circle $x^2+y^2=1$. If this tangent makes an acute angle with the positive $X$-axis in the positive direction, then $a^2 c^2=$

If the focal distance of a point $P\left(2, y_1\right)$ on the parabola $y^2=k x$ is 3 , then the equation of the tangent drawn at $P$ to the given parabola is

Normals are drawn from the point $P(8,0)$ to the parabola $y^2=12 x$. If $\theta$ is the acute angle between two non-horizontal normals among them, then $\tan \theta=$

The normal at a point on the parabola $y^2=4 x$ passes through $(5,0)$. If there are two more normals to this parabola passing through $(5,0)$, then the equation of one of these normals is

The equations of common tangents to the parabola $y^2=16 x$ and the circle $x^2+y^2=8$ are

The equation of the given curve is $x^2-4 x+4 y-8=0$. Match the following

$ \begin{array}{lll} \hline & \text { List I } & \text { List II } \\ \hline \text { (A) } & \text { Focus } & \text { (I) }(4,2) \\ \hline \text { (B) } & \text { Vertex } & \text { (II) }(3,2) \\ \hline \text { (C) } & \begin{array}{l} \text { One end of the } \\ \text { latusrectum } \end{array} & \text { (III) }(2,3) \\ \hline \text { (D) } & \begin{array}{l} \text { point of intersection of the } \\ \text { axis and directrix } \end{array} & \text { (IV) }(2,4) \\ \hline & & \text { (V) }(2,2) \\ \hline \end{array} $

$ \text { The correct match is } $

If one end of a focal chord of the parabola $y^2=\frac{8}{a} \times(a>0)$ is at $(1,4)$, then the length of this focal chord is

If the focal chord drawn through the point $(1,2)$ to the parabola $y^2=8 x$ meets this parabola in $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$, then $x_1+x_2=$

If $\left(2 t^2, 4 t\right)$ is a point on the parabola $y^2=8 x$ such that its focal distance is 3 , then $t=$

If $x^2=8 a y$ is the transformed equation of $x^2-4 y+6 x+15=0$ when the origin is shifted to the point $(\alpha, \beta)$ by translation of axes, then $2 \alpha+8 \beta^2=$

Let $L L^{\prime}$ be the latusrectum and $P Q$ be the focal chord of the parabola $y^2=16 x$. If $P=(1,4)$ and $P, L$ lie in the same quadrant, then $L Q=$

If $P\left(\frac{1}{2}, 4\right)$ and $Q$ are the ends of a focal chord of the parabola $y^2=32 x$ and $S$ is the focus of the parabola, then $S Q=$

If the distance from a variable point $P$ to a fixed point $A(a, 0)$ is equal to the perpendicular distance from $P$ to the line $x+y=0$, then the equation of the locus of $P$ is

The point to which the origin is to be shifted by translation of axes so that the transformed equation of $y^2+4 y+8 x-2=0$ will not contain $y$ term and constant term is

Statement $14 x^2+y^2-4 x y-30 x-50 y+40=0$ is the equation of parabola having $(2,3)$ as its focus and $x+2 y+5=0$ as its directrix.

Statement II The equation of the directrix of the parabola $x^2-4 x+16 y+52=0$ is $y+1=0$

Which of the above statements is (are) true?

The cartesian eql tion of the parabola $x=-2+2 t^2, y=2+4 t$ is

The vertex and the focus of the parabola $2 x^2+5 y-6 x+1=0$ respectively, are

The axis of a parabola is along the line $y=x$ and the distance of its vertex $A$ from $(0,0)$ is $\sqrt{2}$ and that of its focus $S$ from $(0,0)$ is $2 \sqrt{2}$. If $A$ and $S$ lie in first quadrant, then the equation of the parabola in parametric form is

If $y^2=16 x$ is the given parabola, then the point of intersection of the focal chord through the point $(2,2)$ and the double ordinate of length 24 is

Let $P Q$ and $R T$ be two focal chords of the parabola $y^2=16 x$. If $P=(4,8)$ are $R=(16,16)$, then $Q T=$

If all the vertices of an equilateral triangle lie on the parabola $y^2=16 x$ and one of them coincides with the vertex of that parabola, then the length of the side of that triangle is

If $m x-y+c=0$ is a normal at a point $P$ on the parabola $y^2=16 x$ and the focal distance of $P$ is 40 units, then $|c|=$

If $P Q$ is a focal chord of the parabola $y^2=4 x$ with focus $S$ and $P=(4,4)$, then $S Q=$

If the parabola $x^2=4 a y,(a>0)$ makes an intercept of length $\sqrt{40}$ units on the line $y=1+2 x$ then $4 a=$

For the parabola $y=\frac{h^3}{3} x^2+\frac{h^2}{2} x-h+\frac{3}{4 h^3}$, if the equation of directrix is $y=k$, then $k: h$

The equation of the common tangent of the parabolas $x^2=108 y$ and $y^2=32 x$ is

Consider the parabola $y^2+2 x+2 y-3=0$ and match the items of List-I with those of the List-II.

$ \begin{array}{llll} \hline & \text { List-I } & & \text { List-II } \\ \hline \text { A. } & 2 x-5=0 & \text { I. } & \text { Vertex } \\ \hline \text { B. } & \left(\frac{3}{2},-1\right) & \text { II. } & \text { Focus } \\ \hline \text { C. } & y+1=0 & \text { III. } & \text { Equation of directrix } \\ \hline \text { D. } & (2,-1) & \text { IV. } & \text { Equation of the axis } \\ \hline & & \text { V. } & \text { Equation of the Latus rectum } \\ \hline \end{array} $

The normal at a point on the parabola $y^2=4 x$ passes through $(5,0)$. If there are two more normals to this parabola which pass through $(5,0)$, the centroid of the triangle formed by the feet of these three normals is