**Circles: Slope of the Line Segment from Point to Center vs. Slope of the Tangent Line**

Consider a circle with point (a,b) on the circle. Let’s compare the slopes of two lines:

(I) The line connecting the center of the circle and the
point (a,b), and

(II) The slope of the tangent line containing point
(a,b). The tangent line is a line that
touches the circle at point (a,b).

**Case 1: The center is (0, 0) (at the origin)**

The equation of the circle is x^2 + y^2 = r^2, where r is
the radius.

Since (a, b) is on the circle, then a^2 + b^2 = r^2.

(I) Slope of the line connecting the center (0,0) and point
(a,b). Then the slope (labeled s1) is:

s1 = (b – 0)/(a – 0) = b/a

(II) Slope of the tangent line connecting (a,b). The slope at point (a, b) would also be the
slope of the tangent line.

Solving for y,

x^2 + y^2 = r^2

y^2 = r^2 – x^2

y = √(r^2 – x^2)

The derivative is:

dy/dx = 1/2 * (-2 * x) * (r^2 – x^2)^(-1/2)

dy/dx = -x / √(r^2 – x^2)

At point, (a, b), the slope (labeled s2) is:

s2 = -a / √(r^2 – a^2)

Since a^2 + b^2 = r^2,

s2 = -a / √(a^2 + b^2 – a^2)

s2 = -a / √(b^2)

s2 = -a/b

Note: Since s1 = b/a and s2 = -a/b, then s2 = -1/s1.

**Case 2: The is center is at point (m,n) with point (a,b) on the circle.**

The equation of the circle becomes (x – m)^2 + (y – n)^2 =
r^2.

(I) Slope of the line
connecting the center (m, n) and point (a,b).
Then the slope (labeled s1) is:

s1 = (b –n)/(a – m)

(II) Slope of the tangent line connecting (a,b).

Solving for y leads to:

(x – m)^2 + (y – n)^2 = r^2

(y – n)^2 = r^2 – (x -,m)^2

y = √(r^2 – (x – m)^2) + n

Taking the derivative:

dy/dx = -(x – m)/√(r^2 – (x-m)^2)

With the point (a,b):

s2 = -(a – m)/√(r^2 – (a – m)^2)

s2 = -(a – m)/√((a – m)^2 + (b – n)^2 – (a – m)^2)

s2 = -(a – m)/(b – n)

Since s1 = (b – n)/(a – m) and s2 = -(a – m)/(b – n), them
s2 = -1/s1

Conclusion: On a circle
with center (m,n), with the point (a,b) on the circle, the slope of the tangent
line through the point is –(a – m)/(b – n).

Eddie

This blog is property
of Edward Shore, 2016.