Answer to Question #225422 in Calculus for Anuj

Question

Question:

If 𝑎 and 𝑏 are two sides of a right angled hypotenuse 𝑐 and let 𝑝 be the perpendicular from the opposite

vertex on the hypotenuse, show that:

1. (𝛿 𝑝/𝛿𝑎)_b =b³/c³

2. (𝛿𝑝/𝛿𝑎 )_A=b/c

Answer 1

Part 1

From Pythagoras theorem

$c^2=a^2+b^2\\$

From the similarity of triangles

$\frac{p}{a}=\frac{b}{c}$

When we differentiate partially with respect to a

$\frac{\partial p}{\partial a}=\frac{b}{c}$

Cubing each side of the equation and neglecting the smallest differentials we get

$\frac{\partial p}{\partial a}=(\frac{b}{c})^3\\ \frac{\partial p}{\partial a}=\frac{b^3}{c^3}$

Part 2

From Pythagoras theorem

$c^2=a^2+b^2\\$

From the similarity of triangles

$\frac{p}{a}=\frac{b}{c}$

When we differentiate partially with respect to a

$\frac{\partial p}{\partial a}=\frac{b}{c}$