List of top Mathematics Questions on Parabola

The normal drawn to the parabola \[ y^2=8x \] at \((2,4)\) meets the parabola again at the point

If \(a\neq 0\) and the line \[ 2bx+3cy+4d=0 \] passes through the points of intersection of the parabolas \[ y^2=4ax \] and \[ x^2=4ay, \] then

PSQ is a focal chord of the parabola \[ y^2=12x. \] \(A\) and \(B\) are respectively the feet of the perpendiculars drawn from \(P\) and \(Q\) on the directrix of the parabola. If the length of \(AB\) is \(7\sqrt3\) and \[ P=(3t^2,6t), \qquad (0<t<1), \] then \(t=\)

Let \(O\) and \(S\) be the vertex and focus of the parabola \[ y^2=4ax \] respectively and \(x=k\) be its double ordinate of length \(2\sqrt6\,a\). If the line \(x=k\) cuts the \(X\)-axis at \(P\), then the length of the double ordinate drawn through \(O\) to the parabola having \(P\) and \(S\) as vertex and focus is

The vertex of the parabola $2y = -3x^2 + 48x - 200$, is

The coordinates of the focus of the parabola given by the equation \(4y - x^2 + 4x - 12 = 0\) are

If the nearest point on the parabola $y^{2}=4x$ to the given point $P(-1,4)$ is $Q(h,k)$ and $PQ=d$, then $h^{2}+k^{2}+d^{2}=$}

For parabola \(y^2=5x\), normal at P meets x-axis at Q. If PQ subtends \(60^\circ\) at vertex, slope of normal is

Common tangent to circle \(x^2+y^2=4\) and parabola \(y^2=8\sqrt2 x\) is \(y=mx+c\). Then \(\sqrt{m+c^2}=\)

An equilateral triangle OAB is inscribed in the parabola $y^2 = 4x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

If the straight line \[ lx+my+n=0 \] touches the parabola \[ y^2=4ax, \] then

If \(\theta\) is angle between parabolas \[ y^2=108x \] and \[ x^2=32y \] then \[ \cos2\theta= \]

If \(P(t_1)\) and \(Q(t_2)\) are two points on the parabola \[ y^2=7x \] If \(t_1=2\) and \(t_2=-4\), then the length of chord \(PQ\) is

If a straight line \[ y=mx+c \] touches the circle \[ x^2+y^2=4 \] and parabola \[ y^2=4x \] then \(2m^2=\)

The product of the slopes of the non-horizontal normals drawn through the point \[ (6,0) \] to the parabola \[ y^2=8x \] is:

If \[ y=mx+\frac{3}{m} \] is a tangent to the parabola \[ y^2=4ax \] at the point \(P(3,\beta)\), where \(\beta<0\), then the value of \[ 3m-\beta \] is:

The equation of the normal to the parabola \[ y^2=12x \] at the point \((3\lambda^2,6\lambda)\) is

Let the parabola \( y = x^2 + px + q \) passing through the point \( (1, -1) \) be such that the distance between its vertex and the x-axis is minimum. Then the value of \( p^2 + q^2 \) is:

If the focus and vertex of a parabola are at a distance of 3 units and 6 units, respectively, from the origin on the positive x-axis, then the equation of the parabola is

Angle between the parabola \[ y^2 = 4(x - 1) \quad \text{and} \quad x^2 + 4(y - 3) = 0 \quad \text{at the common end of their latus rectum is} \]

The axis of a parabola is $x=0$. If the vertex is at a distance 3 from the origin above the x-axis, the vertex of the parabola is at ________.

The equation of the directrix of the parabola $y^2 + 4y + 4x + 2 = 0$ is ______.

The angle between the tangents drawn from the point (1, 4) to the parabola $y^2 = 4x$, is ______.

If the lines \(\frac{x-1}{2} = \frac{y+1}{k} = \frac{z}{2}\) and \(\frac{x+1}{5} = \frac{y+1}{2} = \frac{z}{k}\) are coplanar, then the equation of the plane containing these lines are