Question

Hi, I am studying for a precalculus quiz and I do not understand this hw problem:

"The tangent line to a circle may be defined as the point that intersects a circle in a single point... If the equation of the circle is x^2+y^2=r^2 and the equation of the tangent line is y=mx+b show that:

a. r^2(1+m^2)=b^2 [Hint: the quadratic equation x^2+(mx+b)^2=r^2 has exactly one solution]
b.the point of tangency is (-r^2m/b,r^2/b)
c. the tangent line is perpendicular to the line containing the circle and the point of tangency

Answer 1

a. For the equation
x^2+(mx+b)^2 = r^2
to have one solution, the discriminant of the quadratic equation
Ax^2 + Bx + C = 0
must be zero
B^2 = 4 AC
A = 1+m^2
B = 2mb
C = b^2-r^2
4m^2b^2 = 4(1+m^2)(b^2-r^2)
m^2b^2 = b^2 +b^2m^2 -r^2 -r^2m^2
b^2 -r^2(1+m^2) = 0

b. x = -B/(2A) = -mb/(1+m^2)
= -mb[r^2/b^2] = -mr^2/b

c. Don't you mean the line containing the CENTER OF the circle and the point of tangency?

Answer 2

Thank you so much. you totally rock