Condition of Tangency between a Line and a Parabola

When a line is tangent to a parabola, it means that the line touches the parabola at only one point. At this point of contact, the gradient of the line is equal to the gradient of the parabola. Therefore, the condition of tangency between a line and a parabola is given by:

l² = 4am

where l is the length of the perpendicular from the focus to the tangent, a is the distance from the focus to the vertex, and m is the gradient of the tangent at the point of contact.

Solution to the Given Problem

Given that the line lx + my = 0 is a tangent to the parabola x² = y, we need to find the condition of tangency.

Step 1: Find the Gradient of the Tangent

Differentiating the equation of the parabola with respect to x, we get:

2x = dy/dx

At the point of contact, the gradient of the tangent is given by the gradient of the line lx + my = 0. Therefore, we have:

m = -l/x

Step 2: Find the Distance from the Focus to the Vertex

The equation of the parabola is given by:

x² = y

This is a vertical parabola with vertex at the origin. Therefore, the distance from the focus to the vertex is given by:

a = 1/4

Step 3: Find the Length of the Perpendicular from the Focus to the Tangent

The equation of the tangent is given by:

lx + my = 0

Therefore, the distance from the focus (0, 1/4) to the tangent is given by:

l = |m(1/4)|/√(1+m²)

Substituting for m, we get:

l = |-l/(4x)|/√(1+l²/x²)

Simplifying, we get:

l² = 4x²/(1+l²/x²)

Step 4: Substitute the Values into the Condition of Tangency Formula

Substituting the values of l and a into the condition of tangency formula, we get:

l² = 4am

Substituting for l² and a, we get:

4x²/(1+l²/x²) = 4/4x

Simplifying, we get:

l² = 4mx

Substituting for m, we get:

l² = -4mn

Therefore, the condition of tangency is given by:

l² = 4mn

Hence, the correct option is D.