2017 AMC 10B Problem 10

Question

The lines with equations

ax-2y=c

and

2x+by=-c

are perpendicular and intersect at

(1, -5).

What is

c?

-13

-8

2

8

13

Rewrite both lines in slope-intercept form

Perpendicular slopes and the point

(1,-5)

determine the parameters

The first equation can be rewritten as

y=\dfrac a2x-\dfrac c2,

and the second as

y=-\dfrac2b x-\dfrac cb.

Because the lines are perpendicular, their slopes multiply to

-1,

so

a=b.

Substituting

(x,y)=(1,-5)

into the two original equations gives

a+10=c

and

2-5a=-c.

Adding these equations yields

12-4a=0,

so

a=3

and

c=13.

Thus, the correct answer is E .

Problem 10 in other years