Task
Suppose A = (a_1, a_2) and B = (b_1, b_2) are two points in the plane, determined by constants a_1, a_2, b_1, b_2. Suppose X = (x_1, x_2) is a third point, determined by the variables x_1 and x_2.
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Write an expression that gives the slope of line AX in terms of a_1, a_2, x_1, x_2. Write an expression that gives the slope of line BX in terms of b_1, b_2, x_1, x_2.
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Write a polynomial equation involving a_1, a_2, b_1, b_2, x_1, x_2 that expresses that the lines AX and BX are perpendicular.
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What geometric figure is the solution set of the equation in b)?
IM Commentary
The purpose of this task is to lead students through an algebraic approach to a well-known result from classical geometry, namely, that a point X is on the circle of diameter AB whenever \angle AXB is a right angle. This consists of translating the right-angle condition into the negative reciprocal slope property of perpendicular lines, followed by a short series of algebraic manipulations (principally, completing the square).
Teachers whose students are seeing this geometric result (which is sometimes referred to as the converse to Thales's Theorem) for the first time may wish to preface this task with a numerical exploration of the phenomena, e.g., having them find explicit points X giving a right \angle AXB for A=(-1,0) and B=(1,0). Teachers looking to reduce the algebraic difficulty could specify explicitly that part (c) can be done by completing the square.