All questions of Co-ordinate Geometry for Class 9 Exam

Understanding the Problem
To find the point of intersection of the two lines given by the equations:
- x + y = 6
- x - y = 2
we will solve these equations simultaneously.
Step 1: Solve the First Equation
From the first equation, we can express y in terms of x:
- y = 6 - x
Step 2: Substitute into the Second Equation
Now, we substitute y in the second equation:
- x - (6 - x) = 2
This simplifies to:
- x - 6 + x = 2
- -6 = 2 (which does not make sense, so we need to simplify correctly)
The correct substitution should be:
- x - y = 2
- Substitute y: x - (6 - x) = 2
- x - 6 + x = 2
- 2x - 6 = 2
- 2x = 8
- x = 4
Step 3: Find y
Now that we have x, we can find y using the first equation:
- y = 6 - x
- y = 6 - 4
- y = 2
Conclusion: Point of Intersection
The point of intersection of the two lines is (4, 2).
Final Answer
Thus, the correct option is:
- a) (4, 2)

11

The perpendicular distance of a point from the y-axis is equal to the x-coordinate of the point. In this case, the x-coordinate of the point (-11, -2) is -11. Therefore, the perpendicular distance from the y-axis is 11.

Understanding Quadrants in Coordinate Geometry
In a Cartesian coordinate system, the plane is divided into four quadrants based on the signs of the x (horizontal) and y (vertical) coordinates.
Coordinate System Overview
- The x-axis runs horizontally: positive to the right and negative to the left.
- The y-axis runs vertically: positive upwards and negative downwards.
Identifying the Quadrants
- Quadrant I: (x > 0, y > 0) - both coordinates are positive.
- Quadrant II: (x < 0,="" y="" /> 0) - x is negative, y is positive.
- Quadrant III: (x < 0,="" y="" />< 0)="" -="" both="" coordinates="" are="" />
- Quadrant IV: (x > 0, y < 0)="" -="" x="" is="" positive,="" y="" is="" />
Analyzing Point P (4, -3)
- The given point P has coordinates (4, -3).
- Here, x = 4 (positive) and y = -3 (negative).
Determining the Quadrant
Since the x-coordinate is positive and the y-coordinate is negative, point P lies in Quadrant IV.
Conclusion
Thus, the correct answer is option 'D', as point P (4, -3) is located in the IV Quadrant where x is positive and y is negative.

Explanation:

Points P and Q with same abscissae:
- Abscissae refers to the x-coordinate of a point on a coordinate plane.
- If points P and Q have the same abscissae, it means they lie on the same vertical line.

Different ordinates:
- Ordinates refer to the y-coordinate of a point on a coordinate plane.
- If points P and Q have different ordinates, it means they are at different heights on the coordinate plane.

Points P and Q will definitely lie on:
- Line parallel to y-axis: Since the points have the same abscissae (x-coordinate) and different ordinates (y-coordinate), they will lie on a line parallel to the y-axis.
- This is because for points to have the same x-coordinate but different y-coordinates, they must be aligned vertically along a line parallel to the y-axis.
Therefore, the correct answer is option 'D) Line parallel to y-axis'.

Origin

The point of intersection of both the axes is called the origin. It is denoted by the coordinates (0,0) in a Cartesian coordinate system. The x-axis and the y-axis are perpendicular to each other at the origin.

Explanation:

The origin is a significant point in a coordinate system as it is used as a reference point to locate other points. It serves as the starting point for measuring distances and determining the position of various points in the coordinate plane.

Significance of the Origin:

1. Reference Point: The origin is used as a reference point to locate other points in the coordinate plane. The distance of any point from the origin can be measured using the coordinates of that point.

2. Quadrant Division: The origin divides the coordinate plane into four quadrants - the first quadrant, second quadrant, third quadrant, and fourth quadrant. These quadrants help in identifying the signs of the coordinates of different points.

3. Symmetry: The origin is a point of symmetry in the coordinate plane. Any point equidistant from the origin in opposite directions lies on the same line passing through the origin.

4. Graphing Functions: When graphing functions, the origin serves as a starting point. The coordinates of other points on the graph are determined relative to the origin.

5. Mathematical Operations: The origin is essential for mathematical operations such as finding the slope of a line or determining the distance between two points. These calculations involve the coordinates of specific points relative to the origin.

Conclusion:

The point of intersection of both the axes is called the origin. It is a reference point in the coordinate plane and has various applications in mathematics and geometry. The origin is denoted by the coordinates (0,0) and serves as the starting point for measuring distances and determining the position of other points in the coordinate plane.

Calculating Distance between Two Points

Explanation:
To calculate the distance between two points, we use the distance formula which is given as:

distance = √((x2 - x1)² + (y2 - y1)²)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Calculation:
Given that the coordinates of the two points are (24, 10) and (-48, 10).

Substituting the values in the distance formula, we get:

distance = √((-48 - 24)² + (10 - 10)²)
= √((-72)² + 0²)
= √(5184)
= 72

Therefore, the distance between the two points is 72.

Hence, the correct answer is option 'A' (72).

Understanding Quadrants in the Cartesian Plane
In the Cartesian coordinate system, the plane is divided into four quadrants:
- Quadrant I (QI): Both x (abscissa) and y (ordinate) are positive.
- Quadrant II (QII): The x (abscissa) is negative and y (ordinate) is positive.
- Quadrant III (QIII): Both x (abscissa) and y (ordinate) are negative.
- Quadrant IV (QIV): The x (abscissa) is positive and y (ordinate) is negative.
Negative Abscissa Implications
When a point has a negative abscissa (x-coordinate), it means the point lies to the left of the y-axis. This situation can only occur in:
- Quadrant II (QII): Here, x is negative, and y is positive.
- Quadrant III (QIII): Here, x is negative, and y is also negative.
Conclusion
Thus, if the abscissa of a point is negative, it will lie in either:
- Quadrant II (QII): Negative x, Positive y
- Quadrant III (QIII): Negative x, Negative y
Therefore, the correct answer is option 'B' (II or III quadrant), as both quadrants accommodate a negative abscissa.

Explanation:

To determine the quadrant in which the point lies, we need to consider the signs of the abscissa (x-coordinate) and ordinate (y-coordinate).

Given:
Line equation: y = 3x + 2

We are looking for a point on this line where the abscissa and ordinate are equal. Let's assume the point is (a, a).

Step 1:
Replace x with a and y with a in the equation of the line to find the value of a.
a = 3a + 2

Step 2:
Simplify the equation by combining like terms.
a - 3a = 2
-2a = 2

Step 3:
Solve for a by dividing both sides of the equation by -2.
a = -2/2
a = -1

Step 4:
Now we have the coordinates of the point: (-1, -1).

Step 5:
To determine the quadrant, we need to consider the signs of the coordinates.
- Since the x-coordinate (-1) is negative, the point lies in either the second or third quadrant.
- Since the y-coordinate (-1) is also negative, the point lies in the third quadrant.

Therefore, the correct answer is option 'C' - the point lies in the third quadrant.

Explanation:

To find the mirror image of a point about the x-axis, we need to reflect the given point across the x-axis. The x-coordinate of the point remains the same, but the y-coordinate changes its sign.

Given point: (4, 3)

To reflect this point across the x-axis, we keep the x-coordinate the same and change the sign of the y-coordinate.

The x-coordinate of the given point is 4. Therefore, the x-coordinate of the mirror image will also be 4.

The y-coordinate of the given point is 3. To find the y-coordinate of the mirror image, we change the sign of 3, which gives us -3.

Hence, the mirror image of the point (4, 3) about the x-axis is (4, -3).

Therefore, the correct answer is option 'A' (4, -3).