All questions of Coordinate Geometry for Interview Preparation Exam

A quadrant because all value postive in first quadrant.

B

Area=1/2[9(3)+3(9)+(-2)(-12)]
=1/2[27+27+24]
=1/2[78]
=39 sq.units
Area=39 sq.units

Explanation:

To find the distance of a point from the y-axis, we need to measure the perpendicular distance from the point to the y-axis.

Given: Point (5, 12)

Steps to find the distance:

Step 1: Identify the coordinates of the point.

The given point is (5, 12), where 5 represents the x-coordinate and 12 represents the y-coordinate.

Step 2: Draw a line perpendicular to the y-axis passing through the point.

We draw a line from the point (5, 12) perpendicular to the y-axis. Let's call the point where the line intersects the y-axis as P.

Step 3: Measure the distance between the point and the y-axis.

The distance between the point (5, 12) and the y-axis is the length of the line segment formed by the point and point P on the y-axis.

Step 4: Calculate the distance using the Pythagorean theorem.

The distance can be calculated using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the distance is the length of the line segment, which is the hypotenuse of a right-angled triangle. The other two sides are the x-coordinate (5) and the y-coordinate (12).

Using the Pythagorean theorem, we can calculate the distance as follows:

Distance = √((5)^2 + (12)^2)
= √(25 + 144)
= √169
= 13

Therefore, the distance of the point (5, 12) from the y-axis is 13 units.

Since none of the given options match the correct answer, it seems there is an error in the options provided. The correct answer should be option 'D' (13 units) instead of option 'A' (5 units).

To find the area of a triangle, we can use the formula for the area of a triangle given its vertices.

The formula for the area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is:
Area = 0.5 * |(x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2))|

Let's calculate the area of the given triangle using this formula.

Given vertices are:
A(1, 1)
B(2, 7)
C(3, 3)

Calculating the area using the formula:
Area = 0.5 * |(1*(7-3) + 2*(3-1) + 3*(1-7))|
= 0.5 * |(1*4 + 2*2 + 3*(-6))|
= 0.5 * |(4 + 4 - 18)|
= 0.5 * |-10|
= 0.5 * 10
= 5

Therefore, the area of the given triangle is 5 square units.

Now let's look at the given options:
a) 2 sq. units
b) 24 sq. units
c) 0 sq. units
d) 12 sq. units

Option c) 0 sq. units is the correct answer because the calculated area of the triangle is not equal to any of the given options. The triangle does have a non-zero area, so options a), b), and d) can be eliminated.

Hence, the correct answer is option c) 0 sq. units.

Explanation:
In mathematics, the coordinates of a point in a two-dimensional plane are represented by two numbers called the abscissa (x-coordinate) and the ordinate (y-coordinate). The abscissa represents the horizontal position of the point, while the ordinate represents the vertical position.

Given that the coordinates of the point are (5, 11), we can determine the abscissa by looking at the first number in the pair, which is 5.

So, the abscissa of the point is 5.

Key Points:
- The abscissa of a point represents its horizontal position in the coordinate plane.
- The abscissa is determined by the first number in the pair of coordinates.
- In this case, the abscissa of the point with coordinates (5, 11) is 5.

Therefore, the correct answer is option C) 5.

 In point (-3, 5), x-coordinate is negative and y-coordinate is positive. So, the point lies in the second quadrant.

Definition of Abscissa:
The abscissa of a point is the distance of the point from the y-axis. It is the x-coordinate of the point in a Cartesian coordinate system. The y-axis is the vertical line on the coordinate plane, and the x-coordinate represents the horizontal distance of a point from this line.

Explanation:
In a Cartesian coordinate system, points are represented by their coordinates (x, y). The x-coordinate represents the horizontal distance of a point from the y-axis, while the y-coordinate represents the vertical distance of a point from the x-axis.

Example:
Consider a point P with coordinates (3, 4). The abscissa of this point is 3, which means it is located 3 units away from the y-axis. The ordinate of this point is 4, indicating it is located 4 units away from the x-axis.

The Y-Axis:
The y-axis is the vertical line on the coordinate plane. It is perpendicular to the x-axis and intersects it at the origin, which is the point (0, 0). The y-axis is used to represent the vertical component of a point's position.

Importance of Abscissa:
The abscissa is essential in determining the position of a point in a coordinate system. It provides information about the horizontal distance of a point from the y-axis. By knowing both the abscissa and ordinate of a point, we can precisely locate it on the coordinate plane.

Summary:
The distance of a point from the y-axis is called the abscissa. It is the x-coordinate of the point in a Cartesian coordinate system. The abscissa is crucial in determining the horizontal position of a point and plays a significant role in locating points on the coordinate plane.

Ordinates of the fourth vertex D are (4, 2).

Concept:
The distance between two points P(x1, y1) and Q(x2, y2) in a plane is given by the formula:
Distance PQ = √((x2 - x1)^2 + (y2 - y1)^2)

Given:
Point A(a, 0) and Point P(x, y) such that the distance between them is a + x.

Calculation:
To find the distance between A and P:
a + x = √((x - a)^2 + y^2)
Squaring both sides:
(a + x)^2 = (x - a)^2 + y^2
a^2 + 2ax + x^2 = x^2 - 2ax + a^2 + y^2
4ax = y^2
Therefore, y^2 = 4ax
Therefore, the correct answer is option C which is 4ax.

Explanation:

To determine the position of point Q(8, 6) with respect to the circle, we need to compare the distance between the center of the circle (origin) and point Q with the radius of the circle.

Step 1: Calculate the distance between the center of the circle (origin) and point Q using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Distance = √((8 - 0)^2 + (6 - 0)^2)
= √(8^2 + 6^2)
= √(64 + 36)
= √100
= 10

Step 2: Compare the distance between the center and point Q with the radius of the circle. Since the center is at the origin, the radius of the circle is the distance between the origin and point P(5, 0).

Radius = √((5 - 0)^2 + (0 - 0)^2)
= √(5^2 + 0^2)
= √25
= 5

Step 3: Compare the distance with the radius:
- If the distance is greater than the radius, point Q lies outside the circle.
- If the distance is equal to the radius, point Q lies on the circle.
- If the distance is less than the radius, point Q lies inside the circle.

In this case, the distance between the center and point Q is 10, which is greater than the radius of the circle (5). Therefore, point Q lies outside the circle.

Conclusion: The point Q(8, 6) lies outside the circle. Therefore, the correct answer is option A) outside.

Given Information:
The vertices of the quadrilateral are (1, 7), (4, 2), (1, 1), and (4, 4).

Explanation:
To determine the type of quadrilateral formed by these vertices, we need to consider the properties of different quadrilaterals.

Square:
A square is a quadrilateral with all sides of equal length and all angles equal to 90 degrees.

Parallelogram:
A parallelogram is a quadrilateral in which opposite sides are parallel.

Rhombus:
A rhombus is a quadrilateral with all sides of equal length.

Rectangle:
A rectangle is a quadrilateral with all angles equal to 90 degrees.

Using Properties:
To determine the type of quadrilateral, we can analyze the given vertices and their properties.

Side Lengths:
Using the distance formula, we can calculate the lengths of the sides of the quadrilateral:
AB = √[(4-1)^2 + (2-7)^2] = √[9 + 25] = √34
BC = √[(4-1)^2 + (4-2)^2] = √[9 + 4] = √13
CD = √[(1-4)^2 + (1-4)^2] = √[9 + 9] = √18
DA = √[(1-4)^2 + (1-7)^2] = √[9 + 36] = √45

Angle Measures:
Using the slope formula, we can calculate the slopes of the sides of the quadrilateral:
mAB = (2-7)/(4-1) = -5/3
mBC = (4-2)/(4-1) = 2/3
mCD = (1-4)/(1-4) = 3/0 (undefined)
mDA = (1-7)/(1-4) = 6/-3 = -2

Analysis:
- The quadrilateral does not have all sides of equal length, so it is not a square or a rhombus.
- The quadrilateral does not have all angles equal to 90 degrees, so it is not a rectangle.
- The quadrilateral does not have opposite sides that are parallel, so it is not a parallelogram.

Conclusion:
Based on the analysis above, the quadrilateral formed by the given vertices is not a square, parallelogram, rhombus, or rectangle. Therefore, the correct answer is option 'A' (None of the above).