Question

Calculus help please. Use implicit differentiation to show that any tangent line at a point P to a circle with center C is perpendicular to the radius CP.

I'm not sure how to do this. Thanks!

Answer 1

perpendicular:

Answer 2

Yep

Answer 3

Circle: \[x^2+y^2=1\]

Answer 4

Use implicit differentiation on that.

Answer 5

and solve for y'

Answer 6

2x + 2yy' = 0
yy' = -x
y' = -x/y

Answer 7

thus, the slope is x/y
and you know that any two lines of a slope are perpendicular if say, fg = -1
Thus the tangent line and the radius are perpendicular.

Answer 8

slope of CP = x/y

Answer 9

Sorry how do i know the slope of cp?

Answer 10

your slope is ur tangent line.

Answer 11

the question says, "with center C", so I would define C= (x0,y0)

\[ (x-x_0)^2 + (y-y_0)^2 = r^2 \]
implicitly differentiate:
\[ 2 (x -x_0) dx + 2(y-y_0) dy = 0 \]
\[ 2(y-y_0) dy = -2(x-x_0) dx \]
\[ \frac{dy}{dx} = -\frac{(x-x_0)}{(y-y_0)} \]

now find the slope of any radius at point (x,y):

change in y over change in x :
\[ \frac{y-y_0}{x-x_0} \]
which is the negative reciprocal of the slope of the tangent

Answer 12

Thank you very much