All questions of Coordinate Geometry for Class 9 Exam

Option 1)

Explanation:

The point (0, b) is a point in the coordinate plane, where the first coordinate (x-coordinate) is 0, and the second coordinate (y-coordinate) is b.

To determine on which axis or quadrant this point lies, we need to examine the values of the coordinates:

- The x-coordinate is 0, which means the point lies on the y-axis.
- The y-coordinate is b, which means the point is at a distance of b units from the origin, in the positive y-direction (since b is positive).

Therefore, the point (0, b) lies on the y-axis.

Option D, which says "y-axis", is the correct answer.

Sorry, but I can't generate a response without more information. Could you please provide the missing details?

A(3, 4) are found in I quadrant because both the cordinates are positive

Explanation:

The statement "If x = y then (x, y) = (y, x)" is a mathematical statement that involves the relationship between two variables, x and y. Let's break down the statement and see why option A is the correct answer.

If x = y:

If x and y are equal, then we can substitute y for x or x for y in any equation involving x and y.

(x, y) = (y, x):

This statement means that the ordered pair (x, y) is equal to the ordered pair (y, x). In other words, the order of the values in the ordered pair does not matter.

Therefore, if x = y, then (x, y) = (y, x). This is because the values in the ordered pair are the same, regardless of which value is listed first.

For example, if x = 3 and y = 3, then (x, y) = (3, 3) and (y, x) = (3, 3). Both ordered pairs are equal, so the statement is true.

Conclusion:

Therefore, option A is the correct answer because the statement "If x = y then (x, y) = (y, x)" is true regardless of the values of x and y as long as x and y are equal.

Explanation:
The Y-axis is the vertical line on a Cartesian plane. It is perpendicular to the X-axis, which is the horizontal line on the same plane. Any point on the Y-axis has an x-coordinate of zero because it lies directly on the Y-axis, which is a line passing through the origin of the plane. Here are some more details:

• Definition of abscissa: The abscissa is the x-coordinate of a point on a Cartesian plane. It is measured from the vertical Y-axis to the point in question.

• Understanding the Y-axis: The Y-axis is a vertical line that runs through the origin of the plane. It consists of all points with an x-coordinate of zero.

• Relationship between the X and Y axes: The X and Y axes intersect at the origin of the plane, which has coordinates (0,0). Any point on the X-axis has a y-coordinate of zero, and any point on the Y-axis has an x-coordinate of zero.

• Importance of the origin: The origin of the plane is the starting point for measuring coordinates. It is where the X and Y axes intersect, and it has coordinates (0,0). All other points on the plane can be located relative to the origin.

• Conclusion: The abscissa or x-coordinate of any point on the Y-axis is zero because the Y-axis consists of all points with an x-coordinate of zero.

Suppose x value is 2 and y value is 1 so 2(2)+1=5

Because the absissa is 3

Explanation:

Quadrants are the four regions in a coordinate plane which are separated by the x-axis and y-axis. These quadrants are numbered from 1 to 4 in a counterclockwise direction. Let's understand how to identify the quadrant in which a point lies:

- Quadrant 1 (Q1): This quadrant is on the right side of the y-axis and above the x-axis. The values of x and y are both positive in this quadrant. So, any point lying in this quadrant has positive values for both x and y coordinates.
- Quadrant 2 (Q2): This quadrant is on the left side of the y-axis and above the x-axis. The value of x is negative and the value of y is positive in this quadrant. So, any point lying in this quadrant has negative x-coordinate and positive y-coordinate.
- Quadrant 3 (Q3): This quadrant is on the left side of the y-axis and below the x-axis. The values of both x and y are negative in this quadrant. So, any point lying in this quadrant has negative values for both x and y coordinates.
- Quadrant 4 (Q4): This quadrant is on the right side of the y-axis and below the x-axis. The value of x is positive and the value of y is negative in this quadrant. So, any point lying in this quadrant has positive x-coordinate and negative y-coordinate.

Now, let's identify in which quadrant the point (3, 2) lies:

- The x-coordinate of the point is 3 which is positive.
- The y-coordinate of the point is 2 which is also positive.

Since both the x and y-coordinates of the point are positive, it lies in the Quadrant 1 (Q1).

Therefore, the correct answer is option 'A' which says that the point (3, 2) belongs to quadrant Q1.

Yes, because when abscissa or coordinate of x - axis is 0 the point lies on y - axis.
therefore, the correct answer is option c.

Quadrant 1st and 4th

Finding the point on y-axis at a distance of 6 units in the positive direction of y-axis

Steps:
1. The point on y-axis is denoted as (0, y), where y is the y-coordinate of the point.
2. Since we want the point to be in the positive direction of y-axis, y should be positive.
3. The distance of the point from the origin is given as 6 units. Using the distance formula, we have:
- distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Here, x1 = 0, y1 = 0 (origin), x2 = 0 (point lies on y-axis), y2 = y
- distance = sqrt((0 - 0)^2 + (y - 0)^2) = sqrt(y^2) = y
- Therefore, y = 6
4. So, the point on y-axis at a distance of 6 units in the positive direction of y-axis is (0, 6).

Answer:
The correct answer is option C) (0, 6).

This is because if it will be not 0 at then the point will fall somewhre else on the graph instead of y- axis.So the co-ordinatemust be (0,Y) to fall on y- axis.

**Explanation:**

To determine the quadrant in which a coordinate point lies, we need to consider the signs of its x and y coordinates.

Let's analyze the given coordinates:

- Coordinate (-2, 3): The x-coordinate is -2 (negative) and the y-coordinate is 3 (positive).
- Coordinate (3, -2): The x-coordinate is 3 (positive) and the y-coordinate is -2 (negative).

Based on these coordinates, we can determine the quadrants as follows:

- Coordinate (-2, 3): Since the x-coordinate is negative and the y-coordinate is positive, this point lies in the **II quadrant**.
- Coordinate (3, -2): Since the x-coordinate is positive and the y-coordinate is negative, this point lies in the **IV quadrant**.

Therefore, the correct statement is option **D: Coordinate (-2, 3) lies in II quadrant whereas (3, -2) lies in IV quadrant**.

**Explanation:**

The equation of the x-axis represents all the points that lie on the x-axis. The x-axis is a horizontal line that intersects the y-axis at the origin (0,0).

The equation of the x-axis can be written in the form y = k, where k is a constant. Since the x-axis is a horizontal line, the value of y remains constant for all values of x.

To find the equation of the x-axis, we need to determine the value of y that remains constant. At every point on the x-axis, the value of y is always 0. Therefore, the equation of the x-axis is y = 0.

**Option C: y = 0**

Option C is correct...

Abscissa is negative and ordinate is positive
so it will lie in 2 quadrant

Because in co-ordinates x axis comes first as (x, y)

It is because the quadrant 3 has both of the x and y as negative numbers

**Explanation:**

The point (0, 9) lies on the positive direction of the y-axis. Let's break down the options and analyze them one by one to understand why the correct answer is option 'B'.

a) In quadrant IV:
Quadrant IV is the bottom right quadrant on the coordinate plane. In this quadrant, both the x and y coordinates are positive. However, since the x-coordinate of the given point is 0, it does not lie in quadrant IV.

b) On the positive direction of y-axis:
The positive direction of the y-axis is the upward direction. Since the y-coordinate of the given point is 9, which is positive, it lies on the positive direction of the y-axis. Therefore, option 'B' is correct.

c) In quadrant III:
Quadrant III is the bottom left quadrant on the coordinate plane. In this quadrant, the x-coordinate is negative, but the y-coordinate is positive. Since the x-coordinate of the given point is 0, it does not lie in quadrant III.

d) On the positive direction of x-axis:
The positive direction of the x-axis is the rightward direction. However, since the x-coordinate of the given point is 0, it does not lie on the positive direction of the x-axis.

Therefore, the correct answer is option 'B' because the point (0, 9) lies on the positive direction of the y-axis.

As it is clear that (7,0) lies in the 1st quadrant it is understood that the positive direction of x-axis is the 1st quadrant. Hence, the answer

Understanding the Circle's Tangency to the X-axis
When a circle is tangent to the x-axis, it means that the circle touches the x-axis at exactly one point. This point is known as the point of tangency.
Circle's Center and Radius
- The center of the circle is given as (2, 3).
- The x-coordinate of the center is 2, and the y-coordinate is 3.
Finding the Point of Tangency
Since the x-axis is the line where y = 0, the point of tangency must have a y-coordinate (ordinate) of 0. This is because the circle touches the x-axis at the height of y = 0.
Determining the Radius
- The distance from the center of the circle at (2, 3) to the x-axis is equal to the y-coordinate of the center, which is 3.
- This distance represents the radius of the circle.
Conclusion: Point of Tangency
- The point of tangency lies directly below the center of the circle at the x-axis level.
- Therefore, since the x-coordinate remains the same (2), the coordinates of the point of tangency must be (2, 0).
Hence, the ordinate (y-coordinate) of the point of tangency is 0.
Correct Answer
The correct answer is option 'B' (0).

Explanation:

Quadrants in the Coordinate Plane:
In the coordinate plane, there are four quadrants labeled as I, II, III, and IV. The quadrants are separated by the x-axis and y-axis. The signs of the x and y coordinates determine in which quadrant a point lies.

Coordinates of the Point:
The point C(-5, -2) has x-coordinate as -5 and y-coordinate as -2.

Identification of Quadrant:
To determine in which quadrant the point C(-5, -2) lies, we need to consider the signs of the x and y coordinates.
- Since the x-coordinate is negative (-5), it means the point lies to the left of the y-axis.
- Since the y-coordinate is negative (-2), it means the point lies below the x-axis.

Location of Point C:
Based on the signs of the coordinates:
- The point C(-5, -2) lies to the left of the y-axis (negative x-coordinate).
- The point C(-5, -2) lies below the x-axis (negative y-coordinate).

Conclusion:
Therefore, the point C(-5, -2) lies in the III Quadrant of the coordinate plane.

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.

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).

The abscissa (x-coordinate) is negative in Quadrants II and III because:- In Quadrant II, the x-values are negative while y-values are positive.- In Quadrant III, both x and y values are negative. Therefore, option B is correct as it includes both quadrants where the abscissa is negative.

Understanding the X-Axis
The x-axis is a fundamental concept in coordinate geometry, serving as a horizontal line in a two-dimensional plane.
Definition of Abscissa
- The abscissa refers to the x-coordinate of a point in a Cartesian coordinate system.
- It indicates the horizontal position of a point.
Points on the X-Axis
- Any point on the x-axis has a y-coordinate of 0.
- Therefore, the coordinates of points on the x-axis can be expressed as (x, 0), where 'x' can be any real number.
Why the Correct Answer is C: Any Number
- Since the abscissa is defined as the x-coordinate, any point on the x-axis can have any value for 'x'.
- For example:
- The point (3, 0) has an abscissa of 3.
- The point (-5, 0) has an abscissa of -5.
- The point (0, 0) has an abscissa of 0.
- This flexibility means that the abscissa of points on the x-axis is not restricted to specific numbers like -1, 0, or 1.
Conclusion
In conclusion, the abscissa of all points on the x-axis is indeed "any number," confirming that option 'C' is correct. Understanding this concept is essential for further studies in mathematics and coordinate geometry.

Explanation:

Distance of a point from the x-axis:
- The distance of a point from the x-axis is the perpendicular distance from the point to the x-axis.
- In this case, the point is (4, 6).

Calculating the distance:
- The y-coordinate of the point (4, 6) is 6.
- Since the x-axis is a horizontal line, the distance of the point from the x-axis is the absolute value of the y-coordinate, which is 6.

Minimum distance:
- The minimum distance of the point (4, 6) from the x-axis is the shortest possible distance, which is the absolute value of the y-coordinate.
- Therefore, the minimum distance of the point (4, 6) from the x-axis is 6.
Therefore, the correct answer is option B) 6.

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).

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.