Coordinate Geometry Practice : Equation of lines

This aptitude question tests your understanding of basic properties of lines, given equation of the lines. The best way to solve such questions is to go from answer choices. This one is a medium level difficulty question. Concepts include properties of concurrent lines, point of intersection of lines, and types of quadrilaterals.

Question

The equations of four lines are given: x + 2y - 3 = 0, 3x + 4y - 7 = 0, 2x + 3y - 4 = 0 and 4x + 5y - 6 = 0. What can be said about the four lines?

1. Concurrent
2. Form sides of a parallelogram
3. Form sides of a square
4. None of these

  Correct Answer      Choice (4). None of these.

Explanatory Answer

Approach : Go from answer choices

Choice 1: The four lines are concurrent

Step 1: Let us find the point of intersection of x + 2y - 3 = 0 and 2x + 3y - 4 = 0
Solving the two equations, we get the point of intersection as (-1, 2).

Step 2: Let us check whether the point (-1, 2) lies on the line 4x + 5y - 6 = 0
Substituting x and y values as -1 and 2 respectively, we get 4(-1) + 5(2) - 6 = 0.
Substituting the values of x and y satisfied the equation and hence the point of intersection lies on this line as well.

Step 3: Let us check whether the point (-1, 2) lies on the line 3x + 4y - 7 = 0 to conclude whether the lines are concurrent.
Substituting x and y values as -1 and 2 respectively, we get 3(-1) + 4(2) - 7 = -2.
Had the point been a part of the line, it would have satisfied the equation and we would have got 0 as the answer.

Because all 4 lines do not pass through the same point, the four lines are NOT concurrent

Choices 2 & 3: Form sides of a parallelogram and form sides of a square

As three of the lines pass through one point they cannot form sides of a quadrilateral.
∴ the four lines CANNOT form sides of either a parallelogram or a square

  The correct answer is Choice (4).