Question

9.3.3 quiz: circles centered at the origin
question 5 of 10
which is the standard equation for a circle centered at the origin with radius
r?
a. $x^{2}+y^{2}=r^{2}$
b. $x^{2}+y^{2}=r$
c. $x + y = r$
d. $x^{2}=y^{2}+r^{2}$

Explanation:

Step1: Recall the distance formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\). For a circle centered at the origin \((0,0)\) with a point \((x,y)\) on the circle and radius \(r\), the distance from \((0,0)\) to \((x,y)\) is \(r\).

Step2: Apply the distance formula

Substitute \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(x,y)\) into the distance formula. We get \(r=\sqrt{(x - 0)^{2}+(y - 0)^{2}}\).

Step3: Square both sides

Squaring both sides of the equation \(r=\sqrt{x^{2}+y^{2}}\) gives \(r^{2}=x^{2}+y^{2}\), which can be rewritten as \(x^{2}+y^{2}=r^{2}\).

Answer:

A. \(x^{2}+y^{2}=r^{2}\)