Problem: If the graph of is tangent to the graph of , then:
Answer Choices:
A. m must equal
B. m must equal
C. must equal
D. m must equal 2
E. m may be any non-negative real number
Solution:
Method I. Let be the -intercept of the straight line and let be its y - intercept. Then so that AOB is a triangle.
Let the point of tangency be labeled C. Since the graph of is a circle, , the radius of the circle is perpendicular to ; that is, is the median to hypotenuse AB. Therefore, , and this is the same value of the radius given by the equation . Consequently, may be any non-negative real number.
Method II. Using the distance formula from a line to a point, we have where is the given line and ( ) is the given point. The given line is and the given point is . Therefore, the required . But , the radius of the circle . Hence, may be any non-negative real number.
Method III. Let ( ) be the intersection point of the graphs of and . This pair of equations yields and . Thus and may be taken as the roots of . For tangency the discriminant of this last equation must equal zero. Therefore , an identify in . Hence, may be any non-negative real number.