Question
Find the equation of a line perpendicular to the line X-2y+3=0 and passing through the point (1, - 2)
Answers
Answered by
GPT-4o mini
To find the equation of a line that is perpendicular to the given line \(X - 2Y + 3 = 0\) and passes through the point \((1, -2)\), we can follow these steps:
1. **Rewrite the Given Line in Slope-Intercept Form:**
The equation of the line is
\[
X - 2Y + 3 = 0.
\]
Rearranging it so that \(Y\) is on one side gives:
\[
2Y = X + 3 \implies Y = \frac{1}{2}X + \frac{3}{2}.
\]
From this, we can identify the slope \(m\) of the given line as \(m = \frac{1}{2}\).
2. **Find the Slope of the Perpendicular Line:**
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope \(m_{\perp}\) of the line we want to find is:
\[
m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{1}{2}} = -2.
\]
3. **Use the Point-Slope Form of the Line Equation:**
The point-slope form of a line's equation is given by:
\[
Y - Y_1 = m(X - X_1),
\]
where \((X_1, Y_1)\) is a point on the line and \(m\) is the slope. Here, the point is \((1, -2)\) and the slope is \(-2\):
\[
Y + 2 = -2(X - 1).
\]
4. **Simplifying the Equation:**
Distributing on the right side, we get:
\[
Y + 2 = -2X + 2.
\]
Now, isolate \(Y\):
\[
Y = -2X + 2 - 2 \implies Y = -2X.
\]
5. **Final Equation:**
The equation of the line that is perpendicular to the line \(X - 2Y + 3 = 0\) and passes through the point \((1, -2)\) is:
\[
Y = -2X.
\]
This is the required equation.
1. **Rewrite the Given Line in Slope-Intercept Form:**
The equation of the line is
\[
X - 2Y + 3 = 0.
\]
Rearranging it so that \(Y\) is on one side gives:
\[
2Y = X + 3 \implies Y = \frac{1}{2}X + \frac{3}{2}.
\]
From this, we can identify the slope \(m\) of the given line as \(m = \frac{1}{2}\).
2. **Find the Slope of the Perpendicular Line:**
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope \(m_{\perp}\) of the line we want to find is:
\[
m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{1}{2}} = -2.
\]
3. **Use the Point-Slope Form of the Line Equation:**
The point-slope form of a line's equation is given by:
\[
Y - Y_1 = m(X - X_1),
\]
where \((X_1, Y_1)\) is a point on the line and \(m\) is the slope. Here, the point is \((1, -2)\) and the slope is \(-2\):
\[
Y + 2 = -2(X - 1).
\]
4. **Simplifying the Equation:**
Distributing on the right side, we get:
\[
Y + 2 = -2X + 2.
\]
Now, isolate \(Y\):
\[
Y = -2X + 2 - 2 \implies Y = -2X.
\]
5. **Final Equation:**
The equation of the line that is perpendicular to the line \(X - 2Y + 3 = 0\) and passes through the point \((1, -2)\) is:
\[
Y = -2X.
\]
This is the required equation.