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Find the value of x for which the points (1,3) , (-2, 9) and (x, -1) are collinear.
- a)-3
- b)3
- c)1
- d)1/2
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Find the value of x for which the points (1,3) , (-2, 9) and (x, -1) a...
Let A(1,3), B(-2,9), and C(x,-1)
For to be points collinear,
x1(y2-y3) + x2(y3-y1) + x3(y1-y2)=0
⇒ 1(9-(-1)) + (-2)(-1-3) + x(3-9)=0
⇒ 1(10)+(-2)(-4)+x(-6)=0
⇒ 10+8-6x=0
⇒ 18 = 6x
⇒ x = 3
For to be points collinear,
x1(y2-y3) + x2(y3-y1) + x3(y1-y2)=0
⇒ 1(9-(-1)) + (-2)(-1-3) + x(3-9)=0
⇒ 1(10)+(-2)(-4)+x(-6)=0
⇒ 10+8-6x=0
⇒ 18 = 6x
⇒ x = 3
Most Upvoted Answer
Find the value of x for which the points (1,3) , (-2, 9) and (x, -1) a...
Understanding Collinearity
To determine the value of x that makes the points (1,3), (-2,9), and (x,-1) collinear, we can use the concept of the area of a triangle formed by these points. If the area is zero, the points are collinear.
Area Formula
The formula for the area A of a triangle formed by three points (x1, y1), (x2, y2), and (x3, y3) is:
A = (1/2) * | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
For our points (1,3), (-2,9), and (x,-1):
- x1 = 1, y1 = 3
- x2 = -2, y2 = 9
- x3 = x, y3 = -1
We set the area A to zero for collinearity:
Calculating the Area
Substituting the values into the area formula:
A = (1/2) * | 1(9 - (-1)) + (-2)(-1 - 3) + x(3 - 9) |
This simplifies to:
A = (1/2) * | 1(10) + (-2)(-4) + x(-6) |
A = (1/2) * | 10 + 8 - 6x |
A = (1/2) * | 18 - 6x |
Setting the area to zero:
| 18 - 6x | = 0
Solving for x
This leads to two equations:
1. 18 - 6x = 0 --> 6x = 18 --> x = 3
2. 18 - 6x = 0 does not need a second equation since it yields the same x value.
Thus, the value of x is:
Final Result
The value of x for which the points are collinear is 3 (option b).
To determine the value of x that makes the points (1,3), (-2,9), and (x,-1) collinear, we can use the concept of the area of a triangle formed by these points. If the area is zero, the points are collinear.
Area Formula
The formula for the area A of a triangle formed by three points (x1, y1), (x2, y2), and (x3, y3) is:
A = (1/2) * | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
For our points (1,3), (-2,9), and (x,-1):
- x1 = 1, y1 = 3
- x2 = -2, y2 = 9
- x3 = x, y3 = -1
We set the area A to zero for collinearity:
Calculating the Area
Substituting the values into the area formula:
A = (1/2) * | 1(9 - (-1)) + (-2)(-1 - 3) + x(3 - 9) |
This simplifies to:
A = (1/2) * | 1(10) + (-2)(-4) + x(-6) |
A = (1/2) * | 10 + 8 - 6x |
A = (1/2) * | 18 - 6x |
Setting the area to zero:
| 18 - 6x | = 0
Solving for x
This leads to two equations:
1. 18 - 6x = 0 --> 6x = 18 --> x = 3
2. 18 - 6x = 0 does not need a second equation since it yields the same x value.
Thus, the value of x is:
Final Result
The value of x for which the points are collinear is 3 (option b).
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Community Answer
Find the value of x for which the points (1,3) , (-2, 9) and (x, -1) a...
Let A(1,3), B(-2,9), and C(x,-1)
For to be points collinear,
x1(y2-y3) + x2(y3-y1) + x3(y1-y2)=0
⇒ 1(9-(-1)) + (-2)(-1-3) + x(3-9)=0
⇒ 1(10)+(-2)(-4)+x(-6)=0
⇒ 10+8-6x=0
⇒ 18 = 6x
⇒ x = 3
For to be points collinear,
x1(y2-y3) + x2(y3-y1) + x3(y1-y2)=0
⇒ 1(9-(-1)) + (-2)(-1-3) + x(3-9)=0
⇒ 1(10)+(-2)(-4)+x(-6)=0
⇒ 10+8-6x=0
⇒ 18 = 6x
⇒ x = 3
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