If at any point on a curve the sub-tangent and sub-normal...

If at any point on a curve the sub-tangent and sub-normal are equal, then the length of the normal is equal to
(A) $\sqrt{2}$ ordinate
(B) ordinate
(C) $\sqrt{2 \text { ordinate }}$
(D) None of these

Key Concepts

Subtangent

The subtangent is the segment on the x-axis that is the projection of the tangent line at a particular point on a curve. Its length is determined by the derivative of the function defining the curve, reflecting the rate at which the curve changes with respect to the x-coordinate.

Subnormal

The subnormal is the segment on the x-axis that is the projection of the normal line (which is perpendicular to the tangent) onto the x-axis at a specific point on the curve. Its length, like that of the subtangent, is related to the derivative of the curve, but it involves a different geometric relationship arising from the perpendicularity condition.

Normal to a Curve

The normal to a curve at a given point is a line perpendicular to the tangent line at that point. The length of the normal segment from the point on the curve to where it intersects a coordinate axis is often studied in related rates and differential geometry problems, helping to understand the geometric properties of the curve.

Ordinate

The ordinate refers to the y-coordinate of a point in the Cartesian coordinate system. In the context of curves, it represents the vertical distance of the point from the x-axis, and it often plays a crucial role in relating various segments like the subtangent, subnormal, and normal with the geometry of the curve.

Differentiation in Curve Analysis

Differentiation is an essential tool in analyzing curves, as it provides the slope of the tangent at any point on the curve. This slope is directly used to determine the lengths of geometric projections such as the subtangent and subnormal. Understanding these relationships requires a grasp of differential calculus and its applications to geometry.

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