If (k, -1), Q(2, 1), and R(4, 5) are collinear, then what is the value of k?
Understand the Problem
The question is asking us to find the value of k such that the points (k, -1), Q(2, 1), and R(4, 5) are collinear. We will use the concept of slope to determine the appropriate value for k.
Answer
The value of \( k \) is \( 1 \).
Answer for screen readers
The value of ( k ) is ( 1 ).
Steps to Solve
- Calculate the slope between points Q and R
We first need to find the slope between the points Q(2, 1) and R(4, 5) using the slope formula:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Substituting the coordinates of Q and R, we get:
$$ m = \frac{5 - 1}{4 - 2} = \frac{4}{2} = 2 $$
- Set up the equation for the slope between points P and Q
Now, we want to find the slope between points P(k, -1) and Q(2, 1). We again use the slope formula:
$$ m_{PQ} = \frac{1 - (-1)}{2 - k} $$
This simplifies to:
$$ m_{PQ} = \frac{2}{2 - k} $$
- Set slopes equal to each other
Since the points are collinear, the slopes must be equal. We set the slopes equal:
$$ \frac{2}{2 - k} = 2 $$
- Solve the equation for k
Now, we cross-multiply to solve for k:
$$ 2 = 2(2 - k) $$
Expanding the right side:
$$ 2 = 4 - 2k $$
Now, isolate k:
$$ 2k = 4 - 2 $$
$$ 2k = 2 $$
$$ k = 1 $$
The value of ( k ) is ( 1 ).
More Information
The points are collinear if they lie on the same straight line. By using the concept of slope, we determined that the point P must have an x-coordinate of 1 for all three points to align perfectly on that line.
Tips
- Not ensuring both slopes are set equal; it's crucial because collinearity requires equal slopes.
- Forgetting to simplify the slope expressions before setting them equal.
