Coordinate Geometry Questions and Answers

Q: Consider the following equations of straight lines:Line L1: \(2x - 3y = 5\)Line L2: \(3x + 2y = 8\)Line L3: \(4x - 6y = 5\)Line L4: \(6x - 9y = 6\)Which one among the following is the correct statement?

The correct statement is that L4 is perpendicular to L2 and L4 is parallel to L3.Write each line as \(y = mx + c\) and compare slopes:L1: \(2x - 3y = 5\), slope \(2/3\)L2: \(3x + 2y = 8\), slope \(-3/2\)L3: \(4x - 6y = 5\), slope \(2/3\)L4: \(6x - 9y = 6\), slope \(2/3\)L1, L3 and L4 are parallel (equal slopes, different intercepts), and each is perpendicular to L2 because \((2/3)(-3/2) = -1\). So L4 is perpendicular to L2 and parallel to L3. The other options each call a parallel pair perpendicular, or a perpendicular pair parallel.

Q: The area of region bounded by the straight lines \(|x| + |y| = 1\) is :

The area is 2 square units.The graph of \(|x| + |y| = 1\) is a square with its corners on the axes at (1, 0), (0, 1), (-1, 0) and (0, -1), a square standing on one corner. Its diagonals both have length 2, and the area of a rhombus or square is half the product of its diagonals:\[ \text{Area} = \tfrac{1}{2} \times 2 \times 2 = 2 \]Equivalently, it is made of four right triangles of legs 1 and 1, each of area 1/2. An answer of 4 comes from taking the square from -1 to 1 on both axes, whose corners lie outside the region, and 1 counts only half of it.

Q: Which of the following will be a point on a line that contains (−1, −1) and (2, 5)?

The point on the line is (100, 201).Find the equation of the line through (−1, −1) and (2, 5).\[ \begin{aligned} m &= \frac{5 - (-1)}{2 - (-1)} = \frac{6}{3} = 2 \\ y - 5 &= 2(x - 2) \\ y &= 2x + 1 \end{aligned} \]For \(x = 100\), \(y = 2(100) + 1 = 201\), so (100, 201) lies on the line.The other points fail the test: for \(x = 30\) the line gives 61, not 41; for \(x = 80\) it gives 161, not 91; and for \(x = 110\) it gives 221, not 121. Those options add 11 to \(x\), a pattern the two given points do not follow.

Important Instructions

71.

Consider the following equations of straight lines:

Line L1: \(2x - 3y = 5\)
Line L2: \(3x + 2y = 8\)
Line L3: \(4x - 6y = 5\)
Line L4: \(6x - 9y = 6\)

Which one among the following is the correct statement?

A.L1 is parallel to L2 and L1 is perpendicular to L3
B.L2 is parallel to L4 and L2 is perpendicular to L1
C.L3 is perpendicular to L4 and L3 is parallel to L2
D.L4 is perpendicular to L2 and L4 is parallel to L3

Exam Year(s)

GATE Life Sciences - 2022, GATE XE - 2022,

Answer: D

The correct statement is that L4 is perpendicular to L2 and L4 is parallel to L3.

Write each line as \(y = mx + c\) and compare slopes:

L1: \(2x - 3y = 5\), slope \(2/3\)
L2: \(3x + 2y = 8\), slope \(-3/2\)
L3: \(4x - 6y = 5\), slope \(2/3\)
L4: \(6x - 9y = 6\), slope \(2/3\)

L1, L3 and L4 are parallel (equal slopes, different intercepts), and each is perpendicular to L2 because \((2/3)(-3/2) = -1\). So L4 is perpendicular to L2 and parallel to L3. The other options each call a parallel pair perpendicular, or a perpendicular pair parallel.

Discuss in Forum

72.

The area of region bounded by the straight lines \(|x| + |y| = 1\) is :

Exam Year(s)

HPPSC HPAS - 2025,

Answer: B

The area is 2 square units.

The graph of \(|x| + |y| = 1\) is a square with its corners on the axes at (1, 0), (0, 1), (-1, 0) and (0, -1), a square standing on one corner. Its diagonals both have length 2, and the area of a rhombus or square is half the product of its diagonals:

\[ \text{Area} = \tfrac{1}{2} \times 2 \times 2 = 2 \]

Equivalently, it is made of four right triangles of legs 1 and 1, each of area 1/2. An answer of 4 comes from taking the square from -1 to 1 on both axes, whose corners lie outside the region, and 1 counts only half of it.

Discuss in Forum

73.

Which of the following will be a point on a line that contains (−1, −1) and (2, 5)?

A.(30, 41)
B.(80, 91)
C.(100, 201)
D.(110, 121)

Exam Year(s)

Kerala PSC KAS - 2020,

Answer: C

The point on the line is (100, 201).

Find the equation of the line through (−1, −1) and (2, 5).

\[ \begin{aligned} m &= \frac{5 - (-1)}{2 - (-1)} = \frac{6}{3} = 2 \\ y - 5 &= 2(x - 2) \\ y &= 2x + 1 \end{aligned} \]

For \(x = 100\), \(y = 2(100) + 1 = 201\), so (100, 201) lies on the line.

The other points fail the test: for \(x = 30\) the line gives 61, not 41; for \(x = 80\) it gives 161, not 91; and for \(x = 110\) it gives 221, not 121. Those options add 11 to \(x\), a pattern the two given points do not follow.

Discuss in Forum

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Tags: Coordinate Geometry Questions and Answers || Coordinate Geometry MCQ || Geometry and Coordinates GK || Distance Formula Quantitative Aptitude || Coordinate Geometry MCQs

Coordinate Geometry Questions and Answers

Important Instructions

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Other Subjects

Tags: Coordinate Geometry Questions and Answers || Coordinate Geometry MCQ || Geometry and Coordinates GK || Distance Formula Quantitative Aptitude || Coordinate Geometry MCQs

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