Parabola and a Line

Equation of chord of the parabola $y^2=16x$ whose midpoint is $\left(1,1\right)$ is

Slope of the chord of parabola $y^2=4ax$ which passes through the point $\left(-6a, 0\right)$ and which subtends an angle of $45°$ at the vertex, is

Let a parabola be $x^2=4y$. Then the image of the parabola formed by the line mirror $x+y=2$, is

The line $y=6x+1$ touches the parabola $y^2=24x$. The coordinates of a point $P$ on this line from which the tangent to $y^2=24x$ is perpendicular to the line $y=6x+1$ is

From the focus of the parabola, $y^2=8x$ as centre, a circle is described so that a common chord of the curves is equidistant from the vertex & focus of the parabola. Find the equation of the circle.

Through the vertex $O$ of the parabola $y^2=4ax,$ variable chords $OP$ and $OQ$ are drawn at right angles. If the variabte chord $PQ$ intersects the axis of $x$ at $R,$ then prove that distance $OR$ is equal to latus rectum of the parabola.

If the line $2y=-4x+11$ is a tangent to the parabola $(y-\beta {)}^2=4(x-\alpha )$ with $x=1$ as directrix, then the value of $\beta$is

An equilateral triangle $\mathrm{ABC}$ is inscribed in the parabola $y=x^2$ and one of the sides of the equilateral triangle has the gradient $2$. If the sum of $x$ -coordinates of all the vertices of the triangle is expressed in the form of $\frac{p}{q}$ (where $p \& q$ are coprime natural number), then the value of  $\left|p-q\right|$ is

A straight line is drawn through the point $P\equiv (3,1)$ at an angle of $60°$ with the positive direction of the $x$- axis to cut the parabola $y^2=8x$ at the points $A$ and $B$. If the value of $|PA-PB|=\frac{a-b}{3},$ then $a+b=$

A line $2 x-y+8=0$ passing through $P\left(-8,-8\right)$ cuts parabola $y=x^2+x-2$ at $A \& B$. Then which is false ?

A line through the origin intersects the parabola $5y=2x^2-9x+10$ at two points whose $x$-coordinates add up to $17$. Then the slope of the line is :

From the point $P\left(\mathrm{4,0}\right)$ the tangents $PA$ and $PB$ are drawn to the circle $x^2+y^2=4.$ If point $Q$ lies on the circle, then the square of the maximum area (in sq. units) of $∆\mathrm{Q}\mathrm{A}\mathrm{B}$ is equal to

The vertices of the base of an isosceles triangle lie on a parabola $y^2=4x$ and the base is a part of the line $y=2x-4$ . If the third vertex of the triangle lies on the $x$ -axis, its coordinates are

If $P$ is a point on the parabola $y^2=8x$ and $A$ is the point $\left(1, 0\right)$, then the locus of the midpoint of the line segment $AP$ is

If $a,b$ and $c$ form a geometric progression with common ratio $r,$ then the sum of the ordinates of the points of intersection of the line $ax+by+c=0$ and the curve $x+2y^2=0$ is

If $P\left(x_1, y_1\right)$ is a point on the ellipse $b^2 x^2+a^2{y}^2=a^2{b}^2$ then the area of $\mathrm{\Delta }SP{S}^{\prime }=$

Chords of an ellipse are drawn through the positive end of the minor axis. Then locus of mid point of chords must be

Equation of chord of the parabola $y^2=16x$ whose mid point is$(1,1)$, is

The mirror image of the directrix of the parabola $y^2=4(x+1)$ in the line mirror $x+2y=3,$ is

In the parabola $y^2=4ax$, the length of the chord passing through the vertex inclined to its axis at $\frac{\pi }{4}$ is