Solve the problem.
-Find an equation for the level curve of the function $f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ that passes through the point $( 3,4 )$.
A) $x ^ { 2 } + y ^ { 2 } = 7$
B) $x + y = 5$
C) $x ^ { 2 } + y ^ { 2 } = 25$
D) $x ^ { 2 } + y ^ { 2 } = 5$

Question

Quizplus · Accepted Answer

A level curve of a function is constructed by finding all the points where the function takes a particular value. Therefore, we want to find values of x and y that make $\sqrt{x^2+y^2}$ equal to a constant k that passes through the point (3, 4). 
Let's call this constant k, so we have:
$$\sqrt{x^2 + y^2} = k$$
Squaring both sides, we get:
$$x^2+y^2=k^2$$
This is the equation of a circle centered at the origin with radius k. 
To find the equation that passes through (3, 4), we need to find the value of k that satisfies this constraint. 
$$\sqrt{3^2+4^2}=\sqrt{25}=5$$
So k=5 and the equation is:
$$x^2+y^2=5^2$$
which is equivalent to choice C.

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