CAT Previous Year Questions| Co Ordinate Geometry

In a parallelogram, opposite sides are parallel, which means that the slopes of opposite sides are equal. We can use this property to find the coordinates of vertex D.

Let's find the slope of side AB, which is the line passing through points A(1, 1) and B(3, 4):

Slope of AB (m_AB) = (y2 - y1) / (x2 - x1) = (4 - 1) / (3 - 1) = 3/2.

Now, since opposite sides of a parallelogram are parallel, the slope of side CD must also be 3/2. We can use this information to find the coordinates of D( x, y).

Let's use point-slope form to find the equation of side CD, knowing the slope (m_CD) and one point (C(-2, 8)):

y - y1 = m_CD(x - x1)
y - 8 = (3/2)(x - (-2))
y - 8 = (3/2)(x + 2)

Now, let's solve for y:

y = (3/2)(x + 2) + 8
y = (3/2)x + 3 + 8
y = (3/2)x + 11

So, the equation of side CD is y = (3/2)x + 11. Now, we can find the x-coordinate of point D by setting y = 4 (the y-coordinate of point B):

4 = (3/2)x + 11

Let's solve for x:

(3/2)x = 4 - 11
(3/2)x = -7

Now, divide by 3/2 to find x:

x = (-7) / (3/2)
x = (-7) * (2/3)
x = -14/3

So, the x-coordinate of point D is -14/3. Now, we can find the y-coordinate of point D using the equation of side CD:

y = (3/2)x + 11
y = (3/2) * (-14/3) + 11
y = -7 + 11
y = 4

Therefore, the coordinates of vertex D are D(-14/3, 4), which is approximately (-4.67, 4).

The closest answer choice to D(-4.67, 4) is option (c) (-4, 5).

To find the value of 'c' in the equation x + 9y + c = 0 for the other two vertices of the parallelogram, you can use the fact that opposite sides of a parallelogram are parallel.

The given points (2, 1) and (-3, -4) are opposite vertices of the parallelogram. So, the line joining these two points is parallel to the line x + 9y + c = 0.

First, let's find the slope of the line passing through the points (2, 1) and (-3, -4):

Slope (m) = (y2 - y1) / (x2 - x1)
Slope (m) = (-4 - 1) / (-3 - 2)
Slope (m) = (-5) / (-5)
Slope (m) = 1

Now, since this line is parallel to the line x + 9y + c = 0, it means that their slopes are equal. So, we have:

1 = the slope of x + 9y + c

Now, we need to find 'c' in the equation x + 9y + c = 0. We know that the coefficient of 'y' in this equation is 9, which is the slope. Therefore, we can write:

9 = 1

Now, solve for 'c':

c = 9 - 1
c = 8

So, the value of 'c' is 8. None of the provided options (15, 12, 13, 14) match the correct value of 'c.'

The area of the closed region bounded is given by the equation,
| x | + | y | = 2.
We can substitute x = 0 or y = 0.
The coordinates we obtain are as follows;
(2,2) , (-2,2) , (2,-2) and (-2,-2)
On joining these points you will get a square whose diagonal is 4 units. Therefore, the sides of the square will be 2√(2) and its area will be 2√(2) × 2√(2) = 8
The area of the closed region is 8 sq. units.

The question is "The area of the closed region bounded by the equation | x | + | y | = 2 in the two-dimensional plane is"

Hence, the answer is 8 sq. units

Choice C is the correct answer.