All questions of Coordinate Geometry for MAT Exam

B

Let A(-2, 2)
B(8, -2)
C(-4, -3)
distance
AB=√(x2-x1) ²+(y2-y1) ²
=√(8+2) ²+(-2-2) ²=√100+16=√116
BC=√(-4-8) ²+(-3+2) ²=√144+1=√145
AC=√(-4+2) ²+(-3-2) ²=√4+25=√29
from the above
AB²=(√116) ²=116
BC²=(√145) ²=145
AC²=(√29) ²=29
AB²+AC²=BC²=145units
hence, by Converse of Pythagoras theorem
ABC is a right triangle
and hence,
A, B, C are the vertices of a right ∆
hence, option (C) is correct

Distance between two point is

Vertices of a Square
The given points (1, 7), (4, 2), (-1, 1), and (-4, 4) can form the vertices of a square.

Properties of a Square
- A square is a special type of rectangle and rhombus.
- All sides of a square are equal in length.
- Opposite sides of a square are parallel.
- All angles in a square are right angles (90 degrees).
- The diagonals of a square are equal in length and bisect each other at right angles.

Verifying the Given Points
To verify if the given points form a square, we need to check the properties mentioned above.
- Calculate the distances between the points to ensure that all sides are equal.
- Check if the slopes of opposite sides are equal to show that they are parallel.
- Calculate the angles between the sides to confirm that they are all right angles.
- Calculate the lengths of the diagonals and check if they are equal.

Conclusion
After verifying the properties of the given points, if all conditions hold true, then the points (1, 7), (4, 2), (-1, 1), and (-4, 4) form the vertices of a square. Hence, the correct answer is option 'D' - square.

1b) 2c) 3d) 4

Solution:
Since the points (p, 0), (0, q) and (1, 1) are collinear, the slope of the line passing through these points is the same. Therefore,

(q-1)/(0-1) = (0-1)/(p-1)

Simplifying this, we get:

q-1 = 1-p

or

p+q=2

Multiplying both sides by 1/pq, we get:

1/q + 1/p = 2/pq

Therefore, the answer is (b) 2.

To determine the value of k, we need to check if the three given points (3, 2), (4, k), and (5, 3) are collinear. Two points are collinear if the slope between them is the same as the slope between any other two points on the line.

Calculating the slope between (3, 2) and (4, k):
The slope formula is given by:

m = (y2 - y1) / (x2 - x1)

Using the coordinates (3, 2) and (4, k), we have:

m = (k - 2) / (4 - 3)
m = (k - 2) / 1
m = k - 2

Calculating the slope between (4, k) and (5, 3):
Using the coordinates (4, k) and (5, 3), we have:

m = (3 - k) / (5 - 4)
m = (3 - k) / 1
m = 3 - k

Since the two slopes are equal, we can equate them:

k - 2 = 3 - k

Solving this equation for k:

2k = 5

k = 5/2

Therefore, the value of k is 5/2, which corresponds to option C.

Let's call the two given vertices A and B, with coordinates (3, 2) and (x, y) respectively.

Since a parallelogram has opposite sides that are parallel, we can find the coordinates of the other two vertices by applying the same translation to points A and B.

To find the translation, we can subtract the x-coordinate of point A from the x-coordinate of point B, and subtract the y-coordinate of point A from the y-coordinate of point B:

x - 3 = 3 - 3 = 0
y - 2 = 2 - 2 = 0

So the translation is (0, 0).

To find the coordinates of the other two vertices, we can add the translation to points A and B:

Point A + translation = (3, 2) + (0, 0) = (3, 2)
Point B + translation = (x, y) + (0, 0) = (x, y)

Therefore, the other two vertices of the parallelogram are (3, 2) and (x, y).

C

Finding the Incentre of a Triangle

To find the incentre of a triangle, we need to find the point where the angle bisectors of the triangle meet. The angle bisectors are the lines that divide the angles of the triangle into two equal parts. The incentre is the intersection of these angle bisectors and is equidistant from all three sides of the triangle.

Given Triangle

The triangle in the question has sides along the lines x = 0, y = 0 and x/6 y/8 = 1. Let's first plot these lines on a graph to get a visual representation of the triangle.

We can see that the triangle is a right-angled triangle with vertices at (0, 0), (6, 0) and (0, 8). Now, let's find the equation of the angle bisectors of this triangle.

Finding the Equation of Angle Bisectors

To find the equation of the angle bisectors, we need to first find the equations of the two lines that form each angle. We can then find the midpoint of these lines and the slope of the angle bisector. Finally, we can use the point-slope form of a line to find the equation of the angle bisector.

Let's start with the angle formed by sides (0, 0) to (6, 0) and (0, 8). The equation of the line joining these two points is y = -3/4x + 8. Now, let's find the equation of the line perpendicular to this line that passes through the midpoint of the line segment joining (0, 0) and (6, 0).

The midpoint of the line segment joining (0, 0) and (6, 0) is (3, 0). The slope of the line joining (0, 0) and (6, 0) is 0. Therefore, the slope of the line perpendicular to this line is undefined (since it is a vertical line). The equation of the line passing through (3, 0) with undefined slope is x = 3.

Using the same method, we can find the equation of the angle bisector of the angle formed by sides (0, 0) to (0, 8) and (6, 0). The equation of the line joining these two points is y = 8/6x = 4/3x. The midpoint of the line segment joining (0, 0) and (0, 8) is (0, 4). The slope of the line joining (0, 0) and (0, 8) is undefined. Therefore, the slope of the line perpendicular to this line is 0. The equation of the line passing through (0, 4) with slope 0 is y = 4.

Intersection of Angle Bisectors

Now that we have the equations of the two angle bisectors, we can find their intersection point, which is the incentre of the triangle. The intersection point is the solution to the system of equations:

x = 3

y = 4

y = 4

The solution to this system of equations is (3, 4), which is option (c). Therefore, the incentre of the triangle whose sides are along the lines x = 0, y = 0 and x/6 y/8 = 1 is