A circle centered at the point (1,1) with radius 5 is defined by (x-1)^2 + (y-1)^2 = 25. Find where the derivative is undefined.?

You can do this without calculus. Draw your circle. Draw vertical tangent lines on the left and right sides of the circle. They touch the circle level with its center where y is 1. There are one radius to the left and right of the center. x = 1 - 5 = -4 and x = 1 + 5 = 6

The points are (-4, 1) and (6, 1)

Given that

(x-1)² + (y-1)² = 25

Radius = 5

Center = (1, 1)

(x-1)² - 25 = -(y-1)²

Differentiation with respect ‘x’

2(x-2) - 0 = -2(y-1) dy/dx

2(x-1) / -2(y-1) = dy/dx

dy/dx = (x-1) / -(y-1)

dy/dx = 0

(x-1) / -(y-1) = 0

x - 1 = 0 → x = 1

now derivative is undefined when

x = 1, y = 1.

2 ( x - 1) + 2(y- 1) dy/dx = 0

dy/dx = - (x - 1) / ( y - 1 )

dy/dx = (1 - x) / (y - 1)

Undefined for y = 1

and

x - 1 = ± 5

x = - 4 , x = 6

Undefined at (- 4 , 1) and (6,1)

(1 ± √25, 1) which is (-4,1) and (6,1)

The tangent to (x-1)^2 + (y-1)^2 = 25 at (x₀,y₀) are (x₀-1)(x-1) + (y₀-1)(y-1) = 25

This is vertical when y₀=1, where (x-1)^2 = 25

y' = (1 -x)/(y -1)

Derivative is undefined where y-1 = 0

(x-1)^2 = 25

x-1 = 5 ; x = 6

-(x-1) = 5 ; x = -4

(6,1) (-4,1) check

That's correct.

You can't have y=1.

And the places on the circle where y=1, are x=-4 and x=6.

Answer:

(-4,1) and (6,1)

P.S. Those are the left and right sides of the circle where a tangent line would be vertical (essentially an infinite slope).