Three lines are given by the equations: ax + by + c = 0 bx + cy + a = 0 cx + ay + b = 0 where a, b and c are non-zero real numbers.
Which one of the following is correct?
- AIf two of the lines are parallel, then all three are parallel.
- BIf two of the lines are parallel, then the third is perpendicular to the other two.
- CIf two of the lines are parallel, then the third is parallel to y = x.
- DIf two of the lines are parallel, then the third is perpendicular to y = x.
- EIf two of the lines are perpendicular, then all three meet at a point.
- FIf two of the lines are perpendicular, then the third is parallel to y = x.
- GIf two of the lines are perpendicular, then the third is perpendicular to y = x.
Show the answer and worked solution
answer · F
- AIf two of the lines are parallel, then all three are parallel.
- BIf two of the lines are parallel, then the third is perpendicular to the other two.
- CIf two of the lines are parallel, then the third is parallel to y = x.
- DIf two of the lines are parallel, then the third is perpendicular to y = x.
- EIf two of the lines are perpendicular, then all three meet at a point.
- FIf two of the lines are perpendicular, then the third is parallel to y = x.
- GIf two of the lines are perpendicular, then the third is perpendicular to y = x.
The gradients are −ab, −bc and −ca. Suppose the first two are perpendicular: (−ab)(−bc) = ac = −1, so a = −c. The remaining line has gradient −ca = −c−c = 1, which is parallel to y = x. The same works for either other pairing by symmetry, since each product of two gradients cancels a letter and leaves a ratio equal to −1.