2017 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 10 Easy

The lines with equations ax-2y=c and 2x+by=-c are perpendicular and intersect at (1,-5). What is c? (2017 AMC 10B Problem, Question#10)

  • A.

    -13

  • B.

    -8

  • C.

    2

  • D.

    8

  • E.

    13

Answer:E

Writing each equation in slope-intercept form, we get y=\dfrac{a}{2}x-\dfrac{1}{2}c and y=-\dfrac{2}{b}x-\dfrac{c}{b}. We observe the slope of each equation is \frac a2 and -\frac 2b, respectively. Because the slope of a line perpendicular to a line with slope m is -\frac 1m, we see that \dfrac{a}{2}=-\dfrac{1}{-\dfrac{2}{b}},because it is given that the two lines are perpendicular. This equation simplifies to a=b. Because (1,-5) is a solution of both equations, we deduce a\times1-2\times(-5)=c and 2\times1+b\times(-5)=-c. Because we know that a=b, the equations reduce to a+10=c and 2-5a=-c. Solving this system of equations, we get c=\rm (E)13.

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