AMU 2026 — Mathematics PYQ

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Question

AMU 2026 — Mathematics PYQ

AMU | Mathematics | 2026

The locus of the lines through the vertex of a cone normal to the tangent planes is called :

Choose the correct answer:

Explanation

1. Core Definition of a Reciprocal Cone

By standard mathematical definition in three-dimensional analytical geometry, if we have a primary cone with a fixed vertex, we can draw a tangent plane at any generating line of this cone.

If we then draw a line passing through the same vertex that is perpendicular (normal) to this tangent plane, that line generates a new path as the tangent plane rolls around the original cone. The collection or locus of all such normal lines forms a secondary cone known precisely as the Reciprocal Cone.

2. Mathematical Derivation

Let the equation of a general cone with its vertex at the origin $(0, 0, 0)$ be represented by the homogeneous quadratic equation:

$a x^{2} + b y^{2} + c z^{2} + 2 f y z + 2 g z x + 2 h x y = 0$

The equation of any tangent plane to this cone at a point or generator line is given by:

$l x + m y + n z = 0$

For this plane to touch the cone, it must satisfy the standard condition of tangency:

$A l^{2} + B m^{2} + C n^{2} + 2 F mn + 2 G n l + 2 H l m = 0$

Where $A, B, C, F, G, H$ are the cofactors of elements $a, b, c, f, g, h$ respectively in the determinant of the matrix representing the cone:

$Δ = a h g am p; h am p; b am p; f am p; g am p; f am p; c$

Now, a line passing through the vertex $(0, 0, 0)$ that is normal to this tangent plane will have direction ratios proportional to the coefficients of the plane, namely $(l, m, n)$ . Thus, the equation of the normal line is:

$\frac{x}{l} = \frac{y}{m} = \frac{z}{n}$

Substituting $l \propto x$ , $m \propto y$ , and $n \propto z$ back into the condition of tangency gives the locus of these normal lines:

$A x^{2} + B y^{2} + C z^{2} + 2 F y z + 2 G z x + 2 H x y = 0$

This derived equation represents a new cone, which is the reciprocal cone of the original one.

Conclusion

The locus of lines drawn through the vertex of a cone perpendicular to its tangent planes generates the Reciprocal cone.
Interestingly, the relationship is mutual: the reciprocal of a reciprocal cone brings you right back to the original cone.

This directly matches option (c).