Parabola — JEE Main PYQs

Let the locus of the centre $(\alpha ,\beta ),\beta >0$, of the circle which touches the circle $x^2+(y-1{)}^2=1$ externally and also touches the x-axis be L. Then the area bounded by L and the line $y=4$ is:

The sum of diameters of the circles that touch (i) the parabola $75x^2=64(5y-3)$ at the point $\left(\frac{8}{5},\frac{6}{5}\right)$ and (ii) the y-axis, is equal to ______.

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

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

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 $P:y^2=4ax,a>0$ be a parabola with focus $S$. Let the tangents to the parabola $P$ make an angle of $\frac{\pi }{4}$ 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 $y^2=4x$ such that it subtends an angle of $\frac{\pi }{2}$ at the point (3, 0 ). Let the line segment PQ be also a focal chord of the ellipse …

Let the normal at the point P on the parabola $y^2=6x$ 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:

A circle of radius 2 unit passes through the vertex and the focus of the parabola $y^2=2x$ and touches the parabola $y={\left(x-\frac{1}{4}\right)}^2+\alpha$, where $\alpha >0$. Then $(4\alpha -8{)}^2$ is equal to ______.

Let the focal chord of the parabola $P:y^2=4x$ along the line $\mathrm{L}:\mathrm{y}=\mathrm{mx}+\mathrm{c},\mathrm{m}>0$ meet the parabola at the points M and N . Let the line L be a tangent to the hyperbola $\mathrm{H}:{\mathrm{x}}^2-{\mathrm{y}}^2=4$. If O is the …

If two tangents drawn from a point $(\alpha ,\beta )$ lying on the ellipse $25x^2+4y^2=1$ to the parabola $y^2=4x$
are such that the slope of one tangent is four times the other, then the value of
$(10\alpha +5{)}^2+{\left(16{\beta }^2+50\right)}^2$ equals ______.

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 $P_1$ be a parabola with vertex $(3,2)$ and focus $(4,4)$ and $P_2$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of $P_2$ is $x+2y=$ ____.

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 $\mathrm{y}={\mathrm{m}}_1\mathrm{x}+{\mathrm{c}}_1$ and $\mathrm{y}={\mathrm{m}}_2\mathrm{x}+{\mathrm{c}}_2,{\mathrm{m}}_1\ne {\mathrm{m}}_2$ are two common tangents of circle $x^2+y^2=2$ and parabola $y^2=x$, then the value of $8\left|m_1{m}_2\right|$ is equal …

Let $\mathrm{x}=2\mathrm{t},\mathrm{y}=\frac{{\mathrm{t}}^2}{3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $\mathrm{SA}\perp \mathrm{BA}$, where A is any point on the conic. If k is the ordinate of the centroid of …

The locus of the mid- point of the line segment joining the focus of the parabola $y^2=4ax$ to a moving point of the parabola, is another parabola whose directrix is:

Let the tangent to the parabola $\mathrm{S}:{\mathrm{y}}^2=2\mathrm{x}$ at the point $\mathrm{P}(2,2)$ meet the x -axis at Q and normal at it meet the parabola S at the point R . Then the area (in sq. units) of the triangle PQR is equal to :

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 ____

Let $P$ be a variable point on the parabola $y=4x^2+1$. Then, the locus of the mid-point of the point $P$ and the foot of the perpendicular drawn from the point $P$ to the line $y=x$ is :

It two tangents from a point $P$ to the parabola $y^2=16(x-3)$ are at right angles, then the locus of point $P$ is:

A tangent is drawn to the parabola $y^2=6x$ which is perpendicular to the line $2x+y=1$. Which of the following points does NOT lie on it ?

A tangent and a normal are drawn at the point $P(2,-4)$ on the parabola $y^2=8x$, which meet the directrix of the parabola at the points $A$ and $B$ respectively. If $Q(a,b)$ is a point such that $AQBP$ is a square, then $2a+b$ is equal to :

A tangent line $L$ is drawn at the point $(2,-4)$ on the parabola $y^2=8x$. If the line $L$ is also tangent to the circle $x^2+y^2=a$, then ' $a$ ' is equal to ____.

The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S(>R)$ respectively from the origin, is :

Let a parabola P be such that its vertex and focus lie on the positive x -axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from $\mathrm{O}(0,0)$ to the parabola P which meet P at $S$ and $R$, then the area (in sq. units) of …

Consider the parabola with vertex $\left(\frac{1}{2},\frac{3}{4}\right)$ and the directrix $y=\frac{1}{2}$. Let P be the point where the parabola meets the line $x=-\frac{1}{2}$. If the normal to the parabola at $P$ intersects the parabola again at the point Q, then $(PQ{)}^2$ i …

If the three normals drawn to the parabola, $y^2=2x$ pass through the point $(a,0)a\ne 0$, then ' $a$ ' must be greater than :

Let $C$ be the locus of the mirror image of a point on the parabola $y^2=4x$ with respect to the line $y=x$. Then the equation of tangent to $C$ at $P(2,1)$ is :

If $y=mx+4$ is a tangent to both the parabolas, $y^2=4x$ and $x^2=2by$, then $b$ is equal to:

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

Let a line $y=mx$ $(m>0)$ intersect the parabola, $y^2=x$ at a point P, other than the origin. Let the tangent to it at P meet the $x$-axis at the point Q. If area $(△OPQ)=4$ sq. units, then $m$ is equal to

If one end of a focal chord $AB$ of the parabola $y^2=8x$ is at $A\left(\frac{1}{2},-2\right)$, then the equation of the tangent to it at $B$ is:

Let P be a point on the parabola, ${\mathrm{y}}^2=12\mathrm{x}$ and N be the foot of the perpendicular drawn from $P$ on the axis of the parabola. A line is now drawn through the mid-point $M$ of $PN$, parallel to its axis which meets the parabola at $Q$. If the $y$ …

Let the latus rectum of the parabola $y^2=4x$ be the common chord to the circles ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ each of them having radius $2\sqrt{5}$. Then, the distance between the centres of the circles ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ is:

If the common tangent to the parabolas $y^2=4x$ and $x^2=4y$ also touches the circle, $x^2+y^2=c^2$, then $c$ is equal to

Let $L_1$ be a tangent to the parabola $y^2=4(x+1)$ and $L_2$ be a tangent to the parabola $y^2=8(x+2)$ such that $L_1$ and $L_2$ intersect at right angles. Then $L_1$ and $L_2$ meet on the straight line:

The tangent to the parabola $y^2=4x$ at the point where it intersects the circle $x^2+y^2=5$ in the first quadrant, passes through the point:

If one end of a focal chord of the parabola, $y^2=16x$ is at ( 1,4 ), then the length of this focal chord is:

If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal, then which one of the following is a valid choice for the ordered triad $(\mathrm{a},\mathrm{b},\mathrm{c})$ ?

If the area of the triangle whose one vertex is at the vertex of the parabola, $y^2+4(x-a^2)=0$ and the other two vertices are the points of intersection of the parabola and $y$-axis, is 250 sq. units, then a value of '$a$' is:

Equation of a common tangent to the parabola ${\mathrm{y}}^2=4\mathrm{x}$ and the hyperbola $\mathrm{xy}=2$ is:

Let $\mathrm{P}(4,-4)$ and $\mathrm{Q}(9,6)$ be two points on the parabola, ${\mathrm{y}}^2=4\mathrm{x}$ and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $△\mathrm{PXQ}$ is maximum. Then this m …

The equation of a common tangent to the curves, $y^2=16x$ and $xy=-4$ is :

The equation of a tangent to the parabola, $x^2=8y$, which makes an angle $\theta$ with the positive direction of $x$-axis, is :

If the tangent to the parabola $y^2=x$ at a point $(\alpha ,\beta ),(\beta >0)$ is also a tangent to the ellipse, $x^2+2y^2=1$, then $\alpha$ is equal to

The area in sq. units) of the smaller of the two circles that touch the parabola, $y^2=4x$ at the points $(1,2)$ and the axis is

Let $A(4,-4)$ and $B(9,6)$ be points on the parabola $y^2=4x$. Let $C$ be chosen on the $\mathrm{arc}AOB$ of the parabola, where $O$ is the origin, such that the area of $△ACB$ is maximum. Then, the area (in sq. units) of …