Coordinate Geometry

Let P be a point on the parabola, y² = 12x and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the y-intercept of the line NQ is ${4 \over 3}$, then :

If P is a point on the parabola y = x² + 4 which is closest to the straight line y = 4x $-$ 1, then the co-ordinates of P are :

If two tangents drawn from a point P to the
parabola y² = 16(x $-$ 3) are at right angles, then the locus of point P is :

Let $O$ be the vertex and $Q$ be any point on the parabola, ${{x^2} = 8y}$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1:3$, then locus of $P$ is :

The area (in sq. units) of an equilateral triangle inscribed in the parabola y² = 8x, with one of its vertices on the vertex of this parabola, is :

The shortest distance between the line x $-$ y = 1 and the curve x² = 2y is :

Let the focal chord of the parabola $\mathrm{P}: y^{2}=4 x$ along the line $\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \mathrm{m}>0$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $\mathrm{H}: x^{2}-y^{2}=4$. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is :

The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the origin, is :

Let $R$ be the focus of the parabola $y^{2}=20 x$ and the line $y=m x+c$ intersect the parabola at two points $P$ and $Q$.

Let the point $G(10,10)$ be the centroid of the triangle $P Q R$. If $c-m=6$, then $(P Q)^{2}$ is :