Question

Consider a circle with a radius of 5 centered at origin. Show that the tangent line at any point is perpendicular to the line segment joining the point with the center of the circle.

Answer 1

what course is this?

Answer 2

calc 1

Answer 3

If you find the slope m of the tangent at point (x0,y0) on the circle, and the radius to point (x0,y0) has slope -1/m then you have proven it.

Answer 4

how do we know it has that slope?

Answer 5

use calculus?

First step, the derivative of a curve gives the slope of a line tangent to the curve.
find the derivative of
x^2 + y^2 = 25
so you can find dy/dx

Answer 6

2x+2y=0 ?

Answer 7

treating both x and y as variables
\[ \frac{d}{dx} x^2 = 2x \frac{dx}{dx}= 2x \]
\[ \frac{d}{dx} y^2 = 2y \frac{dy}{dx} \]

Answer 8

now solve for dy/dx (slope of the tangent)

Answer 9

-x/y

Answer 10

now if x0,y0 is a point on the circle with center 0,0
what is the slope of the line ?

Answer 11

Would it be wrong to put in zero?

Answer 12

I don't follow you.