Parabola practice · Mixed difficulty

Chords of the parabola $y^2=4 x$ touch the hyperbola $x^2-y^2=1$. The locus of the point of intersection of the tangents drawn to the parabola at the extremities of such chords is

If the line $ax + y = c$, touches both the curves ${x^2} + {y^2} = 1$ and ${y^2} = 4\sqrt 2 x$, then $\left| c \right|$ is equal to

Two common tangents to the circle $x^2+y^2=2 a^2$ and parabola $y^2=8ax$ are

The mid-point of the line joining the common points of the line $2x - 3y + 8 = 0$ and ${y^2} = 8x,$ is

The normal chord of a parabola ${y^2} = 4ax$ at $\left( {{x_1},{x_1}} \right)$ subtends a right angle at the

From an external point $P$, pair of tangent lines are drawn to the parabola, $y^2=4 x$. If $\theta_1 \& \theta_2$ are the inclinations of these tangents with the axis of $x$ such that, $\theta_1+\theta_2=\frac{\pi}{4}$, then the locus of $P$ is :

$P_1: y^2=49 x$ and $P_2: x^2=4 a y$ are two parabolas. Equation of a tangent to the parabola $P_1$ at a point where it intersects the parabola $P_2$ is:

The locus of a foot of perpendicular drawn to the tangent of parabola $y^2=4 a x$ from focus, is

The line $l x+m y+n=0$ is a normal to the parabola $y^2=4 a x$, if

For the parabola $y^2=8 x$, the length of the latus-rectum is

If $b$ and $k$ are segments of a focal chord of the parabola $y^2=4 a x$, then $k=$

The focal chords of the parabola $y^2 = 16x$ which are tangent to the circle of radius $r$ and
centre (6,0) are perpendicular, then the radius $r$ of the circle is

If two normals to the parabola $(y-2)^2=-12(x+3)$ intersect at right angles then the chord joining their feet passes through a fixed point whose co-ordinates are:

The equation of the parabola with focus $(a, b)$ and directrix $\frac{x}{a}+\frac{y}{b}=1$ is given by

The equation of the parabola whose focus is $\left( {3, - 4} \right)$ and directrix $6x - 7y + 5 = 0$, is

The vertex of the parabola $3 x-2 y^2-4 y+7=0$ is

The equation of the tangent at a point $P(t)$ where $t$ is any parameter to the parabola $y^2=4 a x$, is

The curve described parametrically by the equations $x=t^2+t+1, y=t^2-t+1$, is

A parabola passing through the point $(-4,-2)$ has its vertex at the origin and $y$-axis as its axis. The latus rectum of the parabola is

The eccentricity of the conic represented by the parametric equation $x=\rho-t+1, y=\rho+t+1$

The equation of the tangents to the parabola $y^2=9 x$ which go through the point $(4,10)$ are

The length of the latusrectum of the parabola $x^2-4 x-8 y+12=0$, is

Locus of the poles of focal chords of a parabola is

If the equation of normal to the parabola $y^2=4 a x$ having slope -2 is given by $y+2 x+k a=0$, then value of $k$.

The locus of the mid-point of the line segment joining a point on the parabola $y^2=4 a x$ and the point of contact of circle drawn on focal distance of the point as diameter with the tangent at the vertex, is

A line $L$ passing through the focus of the parabola ${\left( {y - 2} \right)^2} = 4\left( {x + 1} \right)$ intersects the parabola in two distinct points. If m be the slope of the line $L$, then

The equation of a tangent to the hyperbola $4{x^2} - 5{y^2} = 20$ parallel to the line $x - y = 2$ is

The vertex of the parabola ${y^2} + 6x - 2y + 13 = 0,$ is

If $x+y=k$ is a normal to the parabola $y^2=12 x$, then it touches the parabola

The possible coordinate of the point on $y^2=8 x$, which is closest from $\mathrm{x}^2+(\mathrm{y}+6)^2=1$ is :