Posted by **Allen** on Thursday, October 22, 2009 at 11:17pm.

The tangent to a circle may be defined as the line that intersects the circle in a single point, called the point of tangency. 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

r^2(1 + m^2) = b^2

Hint:

The quadratic equation x^2 + (mx + b)^2 = r^2 has exactly one solution.

- Alg2 -
**Reiny**, Thursday, October 22, 2009 at 11:35pm
sub y = mx + b into x^2 + y^2 = r^2

expanding gave me

x^2 + m^2x^2 + 2bmx + b^2 - r^2 = 0

a^2(1+m^2) + 2bmx + b^2 - r^2 = 0

comparing this to Ax + By + C = 0

A = 1+m^2

B = 2bm

C = b^2 - r^2

we want this quadratic to have only one solution, since there is only one point of contact.

(That was the given hint)

So the discriminant of the quadratic formula must be zero, B^2 - 4AC = 0

(2bm)^2 - 4(1+m^2)(b^2 - r^2) = 0

4b^2m^2 - 4(1+m^2)(b^2 - r^2) = 0

b^2m^2 - (1+m^2)(b^2 - r^2) = 0

b^2m^2 - b^2 + r^2 - b^2m^2 + r^2m^2 = 0

- b^2 + r^2 + r^2m^2 = 0

r^2(1 + m^2) = b^2

as requested.

## Answer this Question

## Related Questions

Math (Pre Cal) - Hi, I am studying for a precalculus quiz and I do not ...

mathematics - The circle x2-2x+y2-4y-4=0 lies on the cartesian plane,question 2....

mathematics - The circle x2-2x+y2-4y-4=0 lies on the cartesian plane,question 2....

mathematics - The circle x2-2x+y2-4y-4=0 lies on the cartesian plane,question 2....

math - From an exteror point P that is h units form a circle of radius r, a ...

homework check - which if the following statements is false? a. The points of a ...

mathematics - The circle x2-2x+y2-4y-4=0 lies on the cartesian plane,question 2....

Math - Circle C has radius 4 inches. Circle D had radius 9 inches. These two ...

Math - From an exterior point P that is h units from a circle of radius r, a ...

GEOMETRY INSCRIBED ANGLES MULTIPLE CHOICE QUESTION - segment BC is tangent to ...