Question
If x + 2y + 3 = 0, x + 2y – 7 = 0 and 2x – y – 4 = 0 form the sides of square, the equation of the fourth side is
- 2x – y – 6 = 0
- 2x + y + 6 = 0
- 2x – y – 14 = 0
- 2x – y + 14 = 0
Hint:
We are given the equations of three sides of a square. We have to find the equation of fourth side of the square. To solve the question we will use properties of a square. All sides of the square are equal. The opposite sides are parallel. The distance between the two opposite sides will be equal to the length of the side of a square.
The correct answer is: 2x – y – 14 = 0
We are given the equation of three sides of a square. They are as follows.
x + 2y + 3 = 0
x + 2y - 7 = 0
2x - y - 4 = 0
Square has opposite sides parallel. And the length of all sides of square is same. So, the distance between the opposite sides will be equal to the length of side of a square.
We will take the first two equations.
x + 2y + 3 = 0
x + 2y - 7 = 0
If we see the co-ordinates of x and y are same. We will rearrange the equation in slope intercept form x = my + c . Here, m is slope and c is the x-intercept
x = -2y - 3
x = -2y + 7
The slope of lines is same as they are parallel.
We can denote their intercept as c1 = 7 and c2 = -3
The distance between parallel lines is given as
Substituting the values we get,
It's same with the other sides
The equation is as follows:
2x - y - 4 = 0
In slope intercept form the equation becomes
y = 2x - 4
So, the slope of line is 2 and y intercept is 4.
The other side will be parallel to this line. It's slope is 2 so, the slope of the other line will be 2. Let the y-intercept of the other line be c. As, y = mx + c. We can write
y = 2x + c
Finding the distance between the lines
We will match the options after substituting the values of c
y = 2x + 6
y - 2x - 6 = 0 which we can write as 2x - y + 6 = 0
y = 2x - 14
y - 2x + 14 = 0 which we can write as 2x - y - 14 = 0
Only option 2x - y - 14 = 0 is there. So, it is the right option.
For such questions, we should know properties of square. We should know the formula to find distance between two parallel lines.
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