Locus Of Normals

The given equation: represents a family of lines all at distance 1 from the point . This is the general form of all lines that are tangent to a circle centered at with radius 1. So: - Center of the circle = - Radius = 1 Equation of circle:

Step 1: Finding the normals at given points The given equation of the parabola is: For points and , we use the normal equation for a parabola: Solving for the locus of intersection of the normals, we get:

Let the given points be A(-3, 1, 2) and B(1, -2, 4). Let P(x, y, z) be a point such that . The direction ratios of AP are , , . The direction ratios of BP are , , . Since AP and BP are perpendicular, the dot product of their direction ratios is zero: This is the equation of a sphere.