Question

Each of these problems (Problems 1-4) is worth four points Definition: Two lines or curves are said to be normal to each other at their point of intersection if they intersect there at right angles or, equivalently, if their tangent lines at the point of intersection are 1. A well-known theorem in geometry states that a line which is tangent to a circle is perpendicular to the radius of the circle at the point of tangency. Use implicit differentiation to confim this for the circle x2 +y2-r' by showing that the line tangent to the circle at the point (a, b)s perpendicular to the radius drawn from (0,0) to (a, b). 2. The graph of x3 + уз 6xy is known as the Folium of Descartes. Write an equation for the line that is tangent to this graph at the point (3,3). (Graph from http://www.history.mcs.st-and.ac.uk/Curves/Foliumd.html) 3. The circles x2 + y2-225 and (x-6)" + (y-17)2-100 intersect at the point (129). Show that they are normal to each other at this point. 4. The circles in Problem 3 intersect at another point as well. Find this point (give its coordinates in fraction form) and determine if the circles are normal to each other there. (This problem's solution is challenging primarily because of its length, rather than its actual degree of difficulty. While it is not strictly necessary that you follow the steps on the back of this page exactly, they should help to guide you through the process in a reasonably efficient way.) Page 1 of 2 (a) Expand the left-hand side of the equation (-6)y 17) 100. Then simplify by combining like terms, and arrange so that one side of the equation only has terms with variables and the other side has only a constant (b) Solve the equation x2 +y2-225 for y2 (not for y). (c) Substitute your result from part (b) into your equation from part (a). Substitute for y2, but not for y. Simplify. You should end up with a linear equation in x and y. (Also, the coefficients will all be negative. To make sign errors less likely as we proceed, multiply both sides by -1.) (d) Solve your equation from part (c) for x and substitute into x2+y 225. Expand. You should now have a quadratic equation in y. (e) Make one side of your equation in part (d) zero and combine like terms. Eliminate fractions and/or decimals by multiplying both sides of the equation by the appropriate constant Divide both sides by the greatest common factor of the coefficients to make them smaller. (f) Solve the equation from part (e) by factoring. (N.B.: Because we know that the circles intersect at the point (12,9), we know that one of the factors is y-9. We're looking for the solution that corresponds to the other factor.) (g) Take your answer from part (1) back to the first part of part (d) to find the corresponding value for x. Then determine whether the two circles are normal to each other at this second point of intersection.

Answer 1

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