How can you prove that 2 equations ...x squared plus y squared equals 25... and a linear equation ...y equals -.75x plus 6.. are tangent to each other algebraically and without a calculator?

Answer 1

The solutions (x,y) to the first equation form the circle centered at 0, with radius 5 (the square root of 25). Solving for y, you can see that it consists of two functions of x: y = sqrt(25-x^2), y = -sqrt(25-x^2). Now you must show that y = -0.75x + 6 is tangent to one of them. That is, show that for one of the curves, at some point x', the slope is -0.75, AND the line intersects the curve at x' (that is, the y-values are equal for the curve and line, at point x'). Compute the derivative of each curve, and find for what x' it is equal to -0.75. (For each derivative, there is an easy integer solution which will work. No calculator needed.) Then check if the line intersects the curve at the point x' you find (for each curve). (It will work for one of the curves.) So you just showed that at point x', one of the curves has slope -0.75, and the line (also with slope -0.75) intersects the curve at x'. Therefore, the line is tangent to the curve at point x'. Hope that helps.