Sunday, May 29, 2016

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

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.





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