Question
If (2, 200) lies on the line f(x)=20x , then what reflection point lies on the line g(x)=log20x ?
Answers
GPT-4o mini
To find the reflection point of \((2, 200)\) across the line \(f(x) = 20x\), we first need to identify the line's slope and use some geometric properties.
1. **Identify the line's equation**:
The line \(f(x) = 20x\) has a slope of 20.
2. **Find the equation of the line perpendicular to \(f(x)\)**:
The slope of a line perpendicular to \(f(x)\) is \(-\frac{1}{20}\). Thus, the equation of the line passing through the point \((2, 200)\) and having this slope is given by:
\[
y - 200 = -\frac{1}{20}(x - 2)
\]
Simplifying this, we get:
\[
y - 200 = -\frac{1}{20}x + \frac{2}{20}
\]
\[
y = -\frac{1}{20}x + 200 + \frac{1}{10}
\]
\[
y = -\frac{1}{20}x + 200.1
\]
3. **Find the intersection point of this perpendicular line with \(f(x)\)**:
To find the intersection, set \(y = 20x\):
\[
20x = -\frac{1}{20}x + 200.1
\]
Multiplying through by 20 to eliminate the fraction:
\[
400x = -x + 4002
\]
\[
401x = 4002
\]
\[
x = \frac{4002}{401} = 10
\]
Now, plug \(x = 10\) back into \(f(x)\) to find \(y\):
\[
y = 20(10) = 200
\]
So, the intersection point is \((10, 200)\).
4. **Find the reflection point**:
The x-coordinate of the reflection point will be symmetric about \(x = 10\). The original point \((2, 200)\) is \(8\) (i.e., \(10 - 2\)) units to the left of \(10\). Therefore, the reflection point will be:
\[
(10 + 8, 200) = (18, 200)
\]
5. **Find if this point lies on line \(g(x) = \log_{20}(x)\)**:
We now check if the reflection point \((18, 200)\) satisfies the equation of \(g(x) = \log_{20}(x)\):
\[
g(18) = \log_{20}(18)
\]
Since \(200 \neq \log_{20}(18)\), the point \((18, 200)\) does not lie on the line \(g(x)\).
However, it seems there was a discrepancy — \(g(18)\) would not equal \(200\). So, let’s find a new point that corresponds to \(g\).
6. **Finding the correct point on \(g(x)\)**:
To find a point that does lie on \(g(x)\), we need to consider what we reflected. As \((2, 200)\) reflected across \(f(x)\) gives us \((18, 200)\) which isn’t on \(g(x)\).
Taking the correct perspective, you can instead find a point \(g(x)\) by directly substituing values for \(x\) values such as \(x=20,1,2 \) etc.
A correct assessment would deem \(g(x)\) does not take typical values as 20 and such, rather focus on properties of logarithms.
**Conclusion**: The reflection point of the point \((2, 200)\) across the line \(f(x) = 20x\) is indeed \((18, 200)\), and additional information must be sought to see if a \(g(x)\)-compliant value exists or not.
1. **Identify the line's equation**:
The line \(f(x) = 20x\) has a slope of 20.
2. **Find the equation of the line perpendicular to \(f(x)\)**:
The slope of a line perpendicular to \(f(x)\) is \(-\frac{1}{20}\). Thus, the equation of the line passing through the point \((2, 200)\) and having this slope is given by:
\[
y - 200 = -\frac{1}{20}(x - 2)
\]
Simplifying this, we get:
\[
y - 200 = -\frac{1}{20}x + \frac{2}{20}
\]
\[
y = -\frac{1}{20}x + 200 + \frac{1}{10}
\]
\[
y = -\frac{1}{20}x + 200.1
\]
3. **Find the intersection point of this perpendicular line with \(f(x)\)**:
To find the intersection, set \(y = 20x\):
\[
20x = -\frac{1}{20}x + 200.1
\]
Multiplying through by 20 to eliminate the fraction:
\[
400x = -x + 4002
\]
\[
401x = 4002
\]
\[
x = \frac{4002}{401} = 10
\]
Now, plug \(x = 10\) back into \(f(x)\) to find \(y\):
\[
y = 20(10) = 200
\]
So, the intersection point is \((10, 200)\).
4. **Find the reflection point**:
The x-coordinate of the reflection point will be symmetric about \(x = 10\). The original point \((2, 200)\) is \(8\) (i.e., \(10 - 2\)) units to the left of \(10\). Therefore, the reflection point will be:
\[
(10 + 8, 200) = (18, 200)
\]
5. **Find if this point lies on line \(g(x) = \log_{20}(x)\)**:
We now check if the reflection point \((18, 200)\) satisfies the equation of \(g(x) = \log_{20}(x)\):
\[
g(18) = \log_{20}(18)
\]
Since \(200 \neq \log_{20}(18)\), the point \((18, 200)\) does not lie on the line \(g(x)\).
However, it seems there was a discrepancy — \(g(18)\) would not equal \(200\). So, let’s find a new point that corresponds to \(g\).
6. **Finding the correct point on \(g(x)\)**:
To find a point that does lie on \(g(x)\), we need to consider what we reflected. As \((2, 200)\) reflected across \(f(x)\) gives us \((18, 200)\) which isn’t on \(g(x)\).
Taking the correct perspective, you can instead find a point \(g(x)\) by directly substituing values for \(x\) values such as \(x=20,1,2 \) etc.
A correct assessment would deem \(g(x)\) does not take typical values as 20 and such, rather focus on properties of logarithms.
**Conclusion**: The reflection point of the point \((2, 200)\) across the line \(f(x) = 20x\) is indeed \((18, 200)\), and additional information must be sought to see if a \(g(x)\)-compliant value exists or not.