Parabola

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 $(10\alpha+5)^2+(16\beta^2+50)^2$ 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 $\mathrm P_1$ be a parabola with vertex $(3,2)$ and focus $(4,4)$ and $\mathrm P_2$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of $\mathrm P_2$ is $x+2y=$ ______.

Let $x=2t$, $y=\dfrac{t^2}3$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $SA\perp BA$, where A is any point on the conic. If k is the ordinate of the centroid of the $\Delta SAB$, then $\displaystyle\lim_{t\to1}k$ is equal to :

The area of the region enclosed between the parabolas $y^2=2x-1$ and $y^2=4x-3$ 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:

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 :

The area of the region bounded by $y^2=8x$ and $y^2=16(3-x)$ is equal to:

A circle of radius 2 unit passes through the vertex and the focus of the parabola $y^2=2x$ and touches the parabola $y=\left(x-\dfrac14\right)^2+\alpha$, where $\alpha>0$. Then $(4\alpha-8)^2$ is equal to ______.

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

Let PQ be a focal chord of the parabola $y^2=4x$ such that it subtends an angle of $\dfrac\pi2$ at the point $(3,0)$. Let the line segment PQ be also a focal chord of the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a^2>b^2$. If $e$ is the eccentricity of the ellipse E, then the value of $\dfrac1{e^2}$ 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 $\dfrac\pi4$ 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 length 6.25 units of the parabola $y^2=4x$. If O is the vertex of the parabola, then 10 times the area (in sq. units) of $\Delta$POQ is equal to ______.

The sum of diameters of the circles that touch (i) the parabola $75x^2=64(5y-3)$ at the point $\left(\dfrac85,\dfrac65\right)$ and (ii) the $y$-axis, is equal to ______.

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

Let the function $f(x)=2x^2-\log_ex$, $x>0$, be decreasing in $(0,a)$ and increasing in $(a,4)$. A tangent to the parabola $y^2=4ax$ at a point P on it passes through the point $(8a,8a-1)$ but does not pass through the point $\left(-\dfrac1a,0\right)$. If the equation of the normal at P is $\dfrac x\alpha+\dfrac y\beta=1$, then $\alpha+\beta$ is equal to ______.

Let P and 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 area of the smaller region enclosed by the curves $y^2=8x+4$ and $x^2+y^2+4\sqrt3x-4=0$ is equal to