The length of subtangent of the reetangular hyperbola...

The length of subtangent of the reetangular hyperbola $x^{2}-y^{2}=a^{2}$ at the point $(\alpha, \sqrt{2} a)$ is
(a) $\sqrt{2} a$
(b) $2 \mathrm{a}$
(c) $\frac{1}{2 a}$
(d) $\frac{a^{3 / 2}}{\sqrt{2}}$.

Key Concepts

Rectangular Hyperbola

A rectangular hyperbola is a specific type of hyperbola in which the asymptotes are perpendicular to each other. It is defined by an equation where the squared terms of the variables have equal and opposite coefficients, showcasing symmetric properties. This concept is fundamental in the study of conic sections and provides a basis for analyzing various geometric properties of the curve.

Subtangent

The subtangent of a curve at a given point refers to the segment of the tangent line that projects onto the horizontal axis between the point of tangency and the point where the tangent intersects the x-axis. It is a geometric measure that connects the derivative (slope) of the curve to the actual length from the point of tangency to the x-axis, providing insights into the curve’s local behaviour.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are intermixed rather than separated explicitly. This method is particularly useful when working with curves defined by implicit relations, as it allows for the calculation of derivatives without needing to solve for one variable in terms of the other.

Application of Derivatives in Geometry

The use of derivatives to compute geometric quantities, such as the subtangent, involves finding the slope of the tangent line at a given point and then using that slope to determine distances and intercepts related to the curve. This approach is a key application of differential calculus in relating algebraic equations to their geometric interpretations.

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