The normal y=m x-2 a m-a m^3 to the parabola y^2=4 a x subtends a right angle at the vertex if (

step 1 I, the question is we need to find dot product of two vectors A and B. Of those three dimensions

The normal y=m x-2 a m-a m^3 to the parabola y^2=4 a x...

Key Concepts

Subtended Angle at a Point

The subtended angle is the angle formed at a specific point (often a vertex) by lines drawn from that point to two other distinct points on a curve. Determining when a chord or line 'subtends' a specific angle at a point, such as a right angle, involves analyzing the geometric relationships and slopes of the connecting lines.

Slope and Perpendicularity

Slope is a measure of the steepness or inclination of a line, and it is fundamental in coordinate geometry. Two lines are perpendicular if and only if the product of their slopes is -1. This condition is key when verifying right angles formed by intersecting lines, such as when a normal line on a parabola subtends a right angle at the vertex.

Parabola

A parabola is a conic section defined by a quadratic relation between its coordinates. It has a unique vertex, a focus, and a directrix, and is symmetric about its axis. The standard form of its equation, such as y² = 4ax, is used to analyze its geometric properties such as tangents, normals, and chords.

Normal to a Curve

The normal to a curve at a given point is the line perpendicular to the tangent at that point. In the context of a parabola, the normal line’s equation is derived using the slope of the tangent, and it is essential for studying reflections, envelopes, and the geometry of intersections between curves and lines.

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