Let the image of parabola $x^2=4 y$, in the line $x-y=1$ be $(y+a)^2=b(x-c)$, $a, b, c \in \mathrm{~N}$. Then $a+b+c$ is equal to

An equilateral triangle OAB is inscribed in the parabola $y^2=4 x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having $A B$ as a diameter from the origin is

Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^2=4 x$ be the curve S . Let P be any point on S . Then the locus of the point, which internally divides OP in the ratio 3 : 1, is :

If the chord joining the points $\mathrm{P}_1\left(x_1, y_1\right)$ and $\mathrm{P}_2\left(x_2, y_2\right)$ on the parabola $y^2=12 x$ subtends a right angle at the vertex of the parabola, then $x_1 x_2-y_1 y_2$ is equal to

Let $y^2 = 12x$ be the parabola with its vertex at $O$. Let $P$ be a point on the parabola and $A$ be a point on the $x$-axis such that $\angle OPA = 90^\circ$. Then the locus of the centroid of such triangles $OPA$ is:

Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16,16)$. If $P(\alpha,\ \beta)$ divides this focal chord internally in the ratio $5:2$, then the minimum value of $\alpha + \beta$ is equal to:

Let O be the vertex of the parabola $x^2=4 y$ and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio $2: 3$ be the conic C . Then the equation of the chord of $C$, which is bisected at the point $(1,2)$, is :

Let O be the vertex of the parabola $y^2=4 x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C , then the length of its latus rectum is :

Let one root of the quadratic equation in $x$ :

$ \left(k^2-15 k+27\right) x^2+9(k-1) x+18=0 $

be twice the other. Then the length of the latus rectum of the parabola $y^2=6 k x$ is equal to:

Let chord PQ of length $3 \sqrt{13}$ of the parabola $y^2=12 x$ be such that the ordinates of points P and Q are in the ratio 1:2. If the chord PQ subtends an angle $\alpha$ at the focus of the parabola, then $\sin \alpha$ is equal to :

Let the directrix of the parabola $\mathrm{P}: y^2=8 x$, cut $x$-axis at the point A . Let $\mathrm{B}(\alpha, \beta), \alpha>1$, be a point on $P$ such that the slope of $A B$ is $3 / 5$. If $B C$ is a focal chord of $P$, then six times the area of $\triangle A B C$ is :

Let the parabola $y = x^2 + px + q$ passing through the point $(1, -1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2 + q^2$ is :

Let P be the parabola, whose focus is $(-2,1)$ and directrix is $2 x+y+2=0$. Then the sum of the ordinates of the points on P, whose abscissa is $-$2, is

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}: y^2=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $A B$, parabola $P$ and the $x$-axis, is :

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2 \sqrt{2}$ units from the origin, respectively. If the point $(1, k)$ lies on the parabola, then a possible value of k is :

Let the focal chord PQ of the parabola $y^2=4 x$ make an angle of $60^{\circ}$ with the positive $x$ axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the $y$-axis at the point $(0, \alpha)$, then $5 \alpha^2$ is equal to:

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then (AB)² is equal to :

Let ABCD be a trapezium whose vertices lie on the parabola $\mathrm{y}^2=4 \mathrm{x}$. Let the sides AD and BC of the trapezium be parallel to $y$-axis. If the diagonal AC is of length $\frac{25}{4}$ and it passes through the point $(1,0)$, then the area of $A B C D$ is

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x+2 y=0$ is $\alpha x^2+\beta y^2-\gamma x y-30 x-60 y+225=0$, then $\alpha+\beta+\gamma$ is equal to :

Let the shortest distance from $(a, 0), a>0$, to the parabola $y^2=4 x$ be 4 . Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :

If the line $3 x-2 y+12=0$ intersects the parabola $4 y=3 x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment AB subtends an angle equal to

Let $\mathrm{P}(4,4 \sqrt{3})$ be a point on the parabola $y^2=4 \mathrm{a} x$ and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :

Let the parabola $y=x^2+\mathrm{p} x-3$, meet the coordinate axes at the points $\mathrm{P}, \mathrm{Q}$ and R . If the circle C with centre at $(-1,-1)$ passes through the points $P, Q$ and $R$, then the area of $\triangle P Q R$ is :

Let $C$ be the circle of minimum area touching the parabola $y=6-x^2$ and the lines $y=\sqrt{3}|x|$. Then, which one of the following points lies on the circle $C$ ?

Let $P Q$ be a chord of the parabola $y^2=12 x$ and the midpoint of $P Q$ be at $(4,1)$. Then, which of the following point lies on the line passing through the points $\mathrm{P}$ and $\mathrm{Q}$ ?

Let $\mathrm{PQ}$ be a focal chord of the parabola $y^{2}=36 x$ of length 100 , making an acute angle with the positive $x$-axis. Let the ordinate of $\mathrm{P}$ be positive and $\mathrm{M}$ be the point on the line segment PQ such that PM : MQ = 3 : 1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line $\mathrm{PQ}$?

Let $\mathrm{A}(0,1), \mathrm{B}(1,1)$ and $\mathrm{C}(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $\mathrm{D}$. If the focus of the parabola $y^{2}=4 \mathrm{ax}$ passing through $\mathrm{D}$ is $(\alpha+\beta \sqrt{2}, 0)$, where $\alpha$ and $\beta$ are rational numbers, then $\frac{\alpha}{\beta^{2}}$ is equal to :

Let $R$ be the focus of the parabola $y^{2}=20 x$ and the line $y=m x+c$ intersect the parabola at two points $P$ and $Q$.

Let the point $G(10,10)$ be the centroid of the triangle $P Q R$. If $c-m=6$, then $(P Q)^{2}$ is :

Let $\mathrm{y}=f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y=-\frac{1}{2}$. Then

$S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$ :

If $\mathrm{P}(\mathrm{h}, \mathrm{k})$ be a point on the parabola $x=4 y^{2}$, which is nearest to the point $\mathrm{Q}(0,33)$, then the distance of $\mathrm{P}$ from the directrix of the parabola $\quad y^{2}=4(x+y)$ is equal to :

If the tangent at a point P on the parabola $y^2=3x$ is parallel to the line $x+2y=1$ and the tangents at the points Q and R on the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ are perpendicular to the line $x-y=2$, then the area of the triangle PQR is :

The equations of two sides of a variable triangle are $x=0$ and $y=3$, and its third side is a tangent to the parabola $y^2=6x$. The locus of its circumcentre is :

The distance of the point $(6,-2\sqrt2)$ from the common tangent $\mathrm{y=mx+c,m > 0}$, of the curves $x=2y^2$ and $x=1+y^2$ is :

The equations of the sides AB and AC of a triangle ABC are $(\lambda+1)x+\lambda y=4$ and $\lambda x+(1-\lambda)y+\lambda=0$ respectively. Its vertex A is on the y-axis and its orthocentre is (1, 2). The length of the tangent from the point C to the part of the parabola $y^2=6x$ in the first quadrant is :

Let a tangent to the curve $\mathrm{y^2=24x}$ meet the curve $xy = 2$ at the points A and B. Then the mid points of such line segments AB lie on a parabola with the :

Let the focal chord of the parabola $\mathrm{P}: y^{2}=4 x$ along the line $\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \mathrm{m}>0$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $\mathrm{H}: x^{2}-y^{2}=4$. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is :

If the tangents drawn at the points $\mathrm{P}$ and $\mathrm{Q}$ on the parabola $y^{2}=2 x-3$ intersect at the point $R(0,1)$, then the orthocentre of the triangle $P Q R$ is :

If the length of the latus rectum of a parabola, whose focus is $(a, a)$ and the tangent at its vertex is $x+y=a$, is 16, then $|a|$ is equal to :

Let $P(a, b)$ be a point on the parabola $y^{2}=8 x$ such that the tangent at $P$ passes through the centre of the circle $x^{2}+y^{2}-10 x-14 y+65=0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of all possible values of $b$. Then the value of $A+B$ is equal to :

Let $\mathrm{P}$ and $\mathrm{Q}$ be any points on the curves $(x-1)^{2}+(y+1)^{2}=1$ and $y=x^{2}$, respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval :

The equation of a common tangent to the parabolas $y=x^{2}$ and $y=-(x-2)^{2}$ is

The tangents at the points $A(1,3)$ and $B(1,-1)$ on the parabola $y^{2}-2 x-2 y=1$ meet at the point $P$. Then the area (in unit ${ }^{2}$ ) of the triangle $P A B$ is :

Let P : y² = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of ${\pi \over 4}$ with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is :

Let PQ be a focal chord of the parabola y² = 4x such that it subtends an angle of ${\pi \over 2}$ at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$. If e is the eccentricity of the ellipse E, then the value of ${1 \over {{e^2}}}$ is equal to :

If vertex of a parabola is (2, $-$1) and the equation of its directrix is 4x $-$ 3y = 21, then the length of its latus rectum is :

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $3x + y - 29 = 0$, is ${x^2} + a{y^2} + bxy + cx + dy + k = 0$, then $a + b + c + d + k$ is equal to :

Let the normal at the point on the parabola y² = 6x pass through the point (5, $-$8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :

If the line $y = 4 + kx,\,k > 0$, is the tangent to the parabola $y = x - {x^2}$ at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :

If $y = {m_1}x + {c_1}$ and $y = {m_2}x + {c_2}$, ${m_1} \ne {m_2}$ are two common tangents of circle ${x^2} + {y^2} = 2$ and parabola y² = x, then the value of $8|{m_1}{m_2}|$ is equal to :

Let $x = 2t$, $y = {{{t^2}} \over 3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA \bot BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta$SAB, then $\mathop {\lim }\limits_{t \to 1} k$ is equal to :

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

Let x² + y² + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x² at (2, 4). Then A + C is equal to ___________.

Statement-2 : If the line, $y = mx + {{\sqrt 5 } \over m}\left( {m \ne 0} \right)$ is their common tangent, then $m$ satiesfies ${m^4} - 3{m^2} + 2 = 0$.