Parabola practice · Mixed difficulty

The chords of contact of the pairs of tangents drawn from each point on the line $2 x+y=4$ to the parabola $y^2=-4 x$ pass through the point

The point of the parabola ${y^2} = 18x$, for which the ordinate is three times the abscissa is

The equation $x^2+4 x y+4 y^2-3 x-6 y-4=0$ represents a

If PSQ is the focal chord of the parabola $y^2=8 x$ such that $S P=6$, then the length of $S Q$ is

One of the points on the parabola ${y^2} = 12x$ with focal distance 12, is

Let $A$ and $B$ be two distinct points on the parabola $y^2=4 x$. If The axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining $A$ and $B$ can be

Two parabolas $C$ and $D$ intersect at two different points, where $C$ is $y=x^2-3$ and $D$ is $y=k x^2$. The intersection at which the $\mathrm{x}$ value is positive is designated point $A$, and $x=a$ at this intersection the tangent line $l$ at $A$ to the curve $D$ intersects curve $C$ at point $B$, other than $A$. If $x$-value of point $B$ is 1 , then a is equal to

The coordinates of the focus of the parabola described parametrically by $x = 5{t^2} + 2,\;y = \;10t\; + \;4$ are

If $y = 2x + 3$ is a tangent to the parabola, ${y^2} = 24\;x,$ then its distance from the parallel normal is

$\min \left[\left(x_1-x_2\right)^2+\left(3+\sqrt{\left(1-x_1^2\right)}-\sqrt{4 x_2}\right)^2\right], \forall$ $x_1, x_2 \in R$, is

If $y+b=m_1(x+a)$ and $y+b=m_2(x+a)$ are two tangents to the parabola $y^2=4 a x$, then

Three distinct normals to the parabola ${y^2} = x$ are drawn through a point $\left( {c,0} \right)$, then

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

The angle between the tangents drawn from the origin to the parabola ${y^2} = 4\;a\left( {x - a} \right),$ is

The set of points on the axis of the parabola $y^2=4 a x$, from which three distinct normals can be drawn to the parabola $y^2=4 a x$, is

The condition that the parabolas $y^2=4 c(x-d)$ and $y^2=4ax$ have a common normal other than $X$-axis $(a>0, c>0)$ is