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Chapter 2: Problem 61

Show that the normal line at any point on the circle \(x^{2}+y^{2}=r^{2}\) passes through the origin.

Short Answer

Expert verified
The normal line at any point on the circle \(x^{2}+y^{2}=r^{2}\) is represented by the equation \(y=mx\), where m is the slope. Since this line passes through the origin (0,0), it validates the given statement.

Step by step solution

01

Identify the point on the circle

Choose any point (x, y) on the circle \(x^{2}+y^{2}=r^{2}\). So, the coordinates of the point are (x, y).
02

Formulate the equation of the normal line

Since we need to show that the normal at any point on the circle passes through the origin, we can say that, the line that passes through origin and any point (x, y) on the circle is the normal. The equation of such normal line with slope \(m = y/x\) can be given by \(y=mx\).
03

Show the normal passes through the origin

As the line is passing through the origin (0,0), by substituting x=0 and y=0 in the equation鈥� the point (0,0) satisfies the equation of the line, proving that the normal at any point on the circle indeed passes through the origin.

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