Draw the graphs of the following . 2x-y-2=0, 4x+ 3y - 24=0 and y+4=0 obtain the vertices of triangle so formed and its area
Answers
EXPLANATION.
Graph of the equations.
⇒ 2x - y - 2 = 0. - - - - - (1).
⇒ 4x + 3y - 24 = 0. - - - - - (2).
⇒ y + 4 = 0. - - - - - (3).
As we know that,
From equation (1), we get.
⇒ 2x - y - 2 = 0. - - - - - (1).
Put the values of x = 0 in the equation, we get.
⇒ 2(0) - y - 2 = 0.
⇒ - y - 2 = 0.
⇒ - y = 2.
⇒ y = - 2.
Their Co-ordinates = (0,-2).
Put the value of y = 0 in the equation, we get.
⇒ 2x - (0) - 2 = 0.
⇒ 2x - 2 = 0.
⇒ 2x = 2.
⇒ x = 1.
Their Co-ordinates = (1,0).
From equation (2), we get.
⇒ 4x + 3y - 24 = 0. - - - - - (2).
Put the values of x = 0 in the equation, we get.
⇒ 4(0) + 3y - 24 = 0.
⇒ 3y - 24 = 0.
⇒ 3y = 24.
⇒ y = 8.
Their Co-ordinates = (0,8).
Put the values of y = 0 in the equation, we get.
⇒ 4x + 3(0) - 24 = 0.
⇒ 4x - 24 = 0.
⇒ 4x = 24.
⇒ x = 6.
Their Co-ordinates = (6,0).
From equation (3), we get.
⇒ y + 4 = 0. - - - - - (3).
⇒ y = - 4.
Their Co-ordinates = (0,-4).
Vertices of triangle formed = (3,4), (0,8), (1,0), (6,0), (0,-2), (-1,-4), (0,-4), (9,-4).
Given :-
2x - y - 2 = 0
4x + 3y - 24 = 0
y + 4 = 0
To Find :-
Vertices
Area
Solution :-
2x - y - 2 = 0 (i)
4x + 3y - 24 = 0 (ii)
y = -4 (iii)
Putting x as 0 in 1
2(0) - y = 2
- y = 2
y = -2
(x,y) = (0,-2)
Putting y as 0 in 1
2x - 0 = 2
2x = 2
x = 2/2
x = 1
(x,y) = (1,0)
Putting x as 0 in 2
4(0) + 3y = 24
0 + 3y = 24
3y = 24
y = 24/3
y = 8
(x,y) = (0,8)
Putting y as 0
4x + 3(0) = 24
4x = 24
x = 24/4
x = 6
(x,y) = (6,0)
y + 4 = 0
y = -4
(x,y) = (0,-4)
Let the triangle be ABC
Multiply Eq 1 by 2
2(2x - y - 2) = 0
4x - 2y - 4 = 0
Subtract 1 and 2
4x + 3y - 24 = 0
4x - 2y - 4 = 0
(-) (+) (+) = 0
5y - 20 = 0
5y = 20
y = 20/5
y = 4
2x - y - 2 = 0
2x - 4 - 2 = 0
2x - 6 = 0
2x = 6
x = 6/2
x = 3
A = 3,4
Put y as -4 in 1
2x - (-4) - 2 = 0
2x + 4 - 2 = 0
2x + 2 = 0
2x = -2
x = -2/2
x = -1
Put y as -4
4x + 3y - 24 = 0
4x + 3(-4) - 24 = 0
4x - 12 - 24 = 0
4x - 36 = 0
4x = 36
x = 36/4
x = 9
B = (9, -4)
Now,
In 1
D = √(x₁ - x₂)² + (y₁ - y₂)²
D = √[((-1) - 9)² + (-4) + 4]²
D = √[(-1 - 9)² + (0)]
D = √100
D = 10
In 2
D = √(3 - 3)² + (4 + 4)²
D = √8²
D = 8
Now
Area = ¹/₂ × 10 × 8
Area = 5 × 8
Area = 40 unit²
