How are l, m, n eliminated?

Replace (l, m, n) by (x, y, z), since the generator passes through the origin

What is the cone and its type?

a^2(y^2 + z^2) = b^2 x^2, a right circular cone about the x-axis

b^2(y^2 + z^2) = a^2 x^2, about the x-axis

a^2(y^2 + z^2) = b^2 x^2, about the y-axis

a^2(y^2 + z^2) = b^2 x, about the x-axis

Short answer

The cone with vertex at origin and base circle y²+z²=b², x=a is a²(y²+z²) = b² x².

Find the equation of the cone whose vertex is the origin and base the circle y²+z² = b², x = a.

Cone. Vertex O(0, 0, 0); base (guiding curve) the circle y² + z² = b², x = a.

A line through the origin with direction ratios (l, m, n): x/l = y/m = z/n. ... (1)

It meets the plane x = a at the point where x/l = a/l = t, so

(a, a m/l, a n/l).

This point lies on the circle y² + z² = b²:

(a m/l)² + (a n/l)² = b²
a² (m² + n²) = b² l²        ... (2)

From (1), m/l = y/x and n/l = z/x, i.e. m : n : l = y : z : x. Put (l, m, n) = (x, y, z) in (2):

a² (y² + z²) = b² x²

Answer: the cone is a²(y² + z²) = b² x², a right circular cone with axis along the x-axis.

✓ Answer checked by Claude Opus 5.5 (high effort)