Q. 71 - Calculus

Question

Let $z=f(x,y)$ be a function of two variables. Prove that if the level curves defined by the equations $f(x,y)=c_1$ and $f(x,y)=c_2$ intersect, then the curves are identical. 

Step-by-Step Solution

Answer

We proved that when $c_1=c_2$, the planes are equal.

Step 1: Given information

We are given a function of two variables z= f(x,y)

Step 2: Explanation

The equation of two variable can be given as 

$ax+by=c_1$ and $ax+by=c_2$

Let the two plane intersect and intersection point be $(x_0,y_0)$

Hence the equation of planes becomes

$a{x}_0+b{y}_0=c_1$ and $a{x}_0+b{y}_0=c_2$

As the right hand side is equal the left hand side also equals Hence we proved that when $when c_1=c_2$

The planes are identical.