Question

A circular pond is modeled by the equation x^2 + y^2= 225. A bridge over the pond is modeled by a segment of the equation x – 7y = –75. What are the coordinates of the points where the bridge meets the edge of the pond?
A (9, 12) and (–12, 9)
B (9, 12) and (12, 9)
C (9, –12) and (–12, –9)
D (–9, 12) and (12, –9)

Answer 1

$x^2+y^2=225 \\ x-7y=-75 \\ \\ \hbox{solve the second equation for x:} \\ x-7y=-75 \ \ \ |+7y \\ x=7y-75 \\ \\ \hbox{substitute 7y-75 for x in the first equation:} \\ (7y-75)^2+y^2=225 \\ 49y^2-1050y+5625+y^2=225 \\ 50y^2-1050y+5625=225 \ \ \ \ \ \ \ \ \ |-225 \\ 50y^2-1050y+5400=0 \ \ \ \ \ \ \ \ \ \ \ \ |\div 50 \\ y^2-21y+108=0 \\ y^2-9y-12y+108=0 \\ y(y-9)-12(y-9)=0 \\ (y-12)(y-9)=0 \\ y-12=0 \ \lor \ y-9=0 \\ y=12 \ \lor \ y=9$

$x=7y-75 \\ x=7 \times 12 -75 \ \lor \ x=7 \times 9-75 \\ x=84-75 \ \lor \ x=63-75 \\ x=9 \ \lor \ x=-12 \\ \\ (x,y)=(9,12) \hbox{ or } (x,y)=(-12,9)$

The points are (9,12) and (-12,9).
The answer is A.