Parabola practice · Easy

Equation $a{x^2} + 2hxy + b{y^2} + 2\mathrm{g}x + 2fy + c = 0\;$
$\left( {abc + 2f\mathrm{g}h - a{f^2} - b{\mathrm{g}^2} - c{h^2} \ne 0} \right)$ represents a parabola, if

The parabola ${y^2} = 4ax$ passes through the point $\left( {2, - 6} \right),$ then the length of its latusrectum is

The locus of the point of intersection of perpendicular tangents to the parabola, is

The parametric coordinates of any point on the parabola ${y^2} = 4ax$ can be

The tangent at $(1,7)$ to the curve $x^2=y-6$ touches the circle $x^2+y^2+16 x+12 y+c=0$ at

The equation of parabola with focus $\left( {0,0} \right)$ and directrix $x + y = 4$, is

$P$ and $Q$ are two distinct points on the parabola, $y^{2}=4 x$, with parameters $t$ and $t_{1}$ respectively. If the normal at $P$ passes through $\mathrm{Q}$, then the minimum value of $\mathrm{t}_{1}^{2}$ is :

The length of the latus rectum of the parabola $9 x^2-6 x+36 y+19=0$ is

If the distances of points $P$ and $Q$ from the focus of a parabola $y^2=4ax$ are 4 and 9 respectively, then the distance of the point of intersection of tangents at $P$ and $Q$ from the focus, is

The focal distance of a point on the parabola $y^2-12 x$ is 4. The abscissa of this point 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 equations of the common tangents to the parabola $y=x^2$ and $y=-(x-2)^2$ are

The value of $c$ for which the line $y=2 x+c$ is a tangent to the parabola $y^2=4 a(x+a)$, is

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

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

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

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