Coordinate Geometry - CAT 2011 Sample Questions

Question

From the following choices what is the equation of a line whose x intercept is half as that of the line 3x + 4y = 12 and y intercept is twice as that of the same line.

1. 3x + 8y = 24
2. 8x + 3y = 24
3. 16x + 3y = 24
4. 3x + y = 6
   

Correct Choice is (4) and Correct Answer 3x + y = 6

Explanatory Answer

Note: To find x intercept in a given equation put y = 0 in that equation and find the value for x (i.e., x-intercept).

Put y = 0 to find the x-intercept in 3x + 4y = 12, i.e., 3x + 4(0) = 12, therefore, x-intercept = 12/3 = 4.

Similarly, to find y intercept in a given equation put x = 0 in that equation and find the value for y (i.e., y-intercept).

Substituting x = 0 to find the y-intercept in 3x + 4y = 12, i.e., 3(0) + 4y = 12, therefore, y-intercept = 12/4 = 3

For the equation under question, the x-intercept is half of the x-intercept of the line 3x + 4y = 12 and is equal to 2
And the y intercept of the new line is twice the y-intercept of 3x + 4y = 12 and is equal to 6.

Now the equation of a line whose x-intercept and y-intercept is given can be written as .

Therefore, equation of the line whose x-intercept is 2 and y-intercept is 6 is .

It can be rewritten as 3x + y = 6.

Question

What is the equation of the line that is parallel to the line 3x + 7y = 10 and passes through the point (4, 8)?

1. 7x – 3y = 46
2. 3x + 7y = 44
3. 9x + 21y – 184 = 0
4. 3x + 7y = 68
   

Correct Choice is (4) and Correct Answer 3x + 7y = 68

Explanatory Answer

Note: If two lines are parallel, then their slopes will be the same.

Therefore, if two lines are parallel and the equation of one of the lines is a₁x + b₁y = c₁, then the equation of the line parallel to the line will be of the form a₁x + b₁y = c₂

In this example, the equation of the first line is 3x + 7y = 10
So, the second line will have a generalized equation of the form 3x + 7y = k.

As, the second line passes through the point (4, 8), substituting x = 4 and y = 8 should satisfy the equation of the line.

i.e. 3(4) + 7(8) = k

12 + 56 + = k 0

=> k = 68

Hence, the equation of the line 3x + 7y = 68