Math CentralQuandaries & Queries


Question from Rita, a student:

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


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


I am sure there are many ways to solve this problem. What I did was to use what I know about the sum and product of the roots of a quadratic polynomial.

For the polynomial ax2 + bx + c the sum of the roots is -b/a and the product of the roots is c/a.

If there is only one root, call it k, then 2k = - b/a and k2 = c/a and hence [-b/(2a)]2 = c/a.

Apply this to your quadratic polynomial and see if you cab derive the expression r2(1 + m2) = b2.


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