Find the equations of the tangents to the following circles at the point indicated: x^2+y^2=2x ; at (1,1) ; at (0,0) ; at a point whose abscissa is 1 1/2

Answer 1

x^2 + y^2 = 2x

2x + 2yy' =2

y' = (1-x)/y = slope of the tangent line

at (1,1) slope = 0 tangent line is y=1

at (0,0) the slope is undefined or infinite. the tangent line is vertical, x=0

x^2 + y^2 = 2x

add 1-2x to both sides to get

x^2-2x+1 + y^2 = 1

that's completing the square

(x-1)^2 + y^2 = 1 is a circle with radius 1 and center (1,0)

(1,1) is the top of the circle (0,0) is the left side of the circle