Q. 70 - Calculus

Question

Let $z = f(x, y)$ be a function of two variables. Prove that when $c_1=c_2$ the level curves defined by the equations $f(x,y)=c_1$ and $f(x,y)=c_2$ do not intersect. 

Step-by-Step Solution

Answer

By contradiction we proved that the planes do not intersect

Step 1: Given information.

We are given z= f(x,y)

Step 2: Explanation.

The equation of the two variables can be given as

$ax+by=c_1$ and $ax+by=c_2$

Suppose these two planes intersect ANd the intersection point be $(x_0,y_0)$

Hence the equation becomes

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

As the left-hand side of the equations is the same the right-hand side should be equal,

Hence we get, $c_1=c_2$

But we are given that $c_1\ne c_2$

Hence our assumption is wrong the planes do not intersect