JEE Main & Advanced Mathematics Conic Sections Co-normal Points

The points on the curve at which the normals pass through a common point are called co-normal points. Q, R, S are co-normal points. The co- normal points are also called the feet of the normals.

Properties of co-normal points

(1) Three normals can be drawn from a point to a parabola.

(2) The algebraic sum of the slopes of three concurrent normals is zero.

(3) The sum of the ordinates of the co-normal points is zero.

(4) The centroid of the triangle formed by the co-normal points lies on the axis of the parabola.

(5) The centroid of a triangle formed by joining the foots of the normal of the parabola lies on its axis and is given by

\[\left( \frac{am_{1}^{2}+am_{2}^{2}+am_{3}^{2}}{3},\frac{2a{{m}_{1}}+2a{{m}_{2}}+2a{{m}_{3}}}{3} \right)\]\[=\left( \frac{am_{1}^{2}+am_{2}^{2}+am_{3}^{2}}{3},0 \right)\].

(6) If three normals drawn to any parabola \[{{y}^{2}}=4ax\]from a given point (h, k) be real, then \[h>2a\] for \[a=1\], normals drawn to the parabola \[{{y}^{2}}=4x\] from any point (h, k) are real, if \[h>2\].

(7) Out of these three at least one is real, as imaginary normals will always occur in pairs.