The matrix (5k 0)
(0 k-2) represents an enlargement with the origin as source. Find k
To find the value of k in the given matrix, we can compare the coordinates of a known point before and after the enlargement.
Let's take the point (1, 0) as our known point.
Before the enlargement:
(1, 0)
After the enlargement:
(5k, 0)
Since the enlargement is centered at the origin, we can see that the x-coordinate of the point is multiplied by 5k, and the y-coordinate remains the same.
So comparing the coordinates, we can write:
4k = 1
Solving this equation, we get:
k = 1/4 = 0.25
To determine the value of k in the matrix, you need to find the scale factor of the enlargement.
An enlargement is a transformation that increases or decreases the size of an object. In this case, since the origin (0, 0) is the source, it means that the origin is the center point of the enlargement.
Let's analyze the matrix:
(5k 0)
(0 k-2)
The diagonal elements of the matrix represent the scaling factors along the x-axis and y-axis, respectively. Since the matrix is diagonal, the x-axis and y-axis scaling factors are independent.
For the x-axis scaling factor, we have 5k. This means that any value multiplied by 5k will be stretched or compressed along the x-axis by a factor of 5k.
For the y-axis scaling factor, we have k-2. Similarly, any value multiplied by k-2 will be stretched or compressed along the y-axis by a factor of k-2.
Since we want to find the scale factor of the enlargement, we need to determine the value of k that makes the scaling factors equal.
Setting the x-axis scaling factor equal to the y-axis scaling factor, we have:
5k = k-2
Simplifying, we get:
4k = -2
Dividing both sides by 4, we find:
k = -2/4
Simplifying further, we get:
k = -1/2
Therefore, the value of k is -1/2, which represents the scale factor of the enlargement.
To find the value of k in the matrix (5k 0)
(0 k-2)
representing an enlargement with the origin as the source, we can look at the scaling factors for the x-axis and the y-axis separately.
For the x-axis scaling factor, we look at the value in the (1,1) position of the matrix. In this case, it is 5k. Since the x-axis scaling factor is the ratio of the new x-coordinate to the original x-coordinate, we can set up the equation:
new x-coordinate / original x-coordinate = x-axis scaling factor
Since the origin is the source, the original x-coordinate is 0, and the new x-coordinate is 5k. Plugging in these values, we get:
5k / 0 = 5k
Since this equation doesn't work for any value of k, we can conclude that there is no x-axis scaling factor, meaning the matrix does not represent an enlargement in the x-axis direction.
For the y-axis scaling factor, we look at the value in the (2,2) position of the matrix. In this case, it is k-2. Using the same reasoning as before, we can set up the equation:
new y-coordinate / original y-coordinate = y-axis scaling factor
Since the origin is the source, the original y-coordinate is 0, and the new y-coordinate is (k-2). Plugging in these values, we get:
(k-2) / 0 = undefined
This equation results in an undefined value for any value of k.
Therefore, there is no value of k that satisfies the conditions for an enlargement representation with the origin as the source in this matrix.