Question:
Given the line equation \( Ax + By + C = 0 \), which passes through the point \( (-10, 7) \) and is perpendicular to the line \( 11x - 8y - 16 = 0 \), find \( C \).
Given the line equation \( Ax + By + C = 0 \), which passes through the point \( (-10, 7) \) and is perpendicular to the line \( 11x - 8y - 16 = 0 \), find \( C \).
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For perpendicular lines, use the property that the product of their slopes equals \( -1 \). Find the slope of the given line and use its negative reciprocal to determine the slope of the perpendicular line.
Updated On: Apr 24, 2025
- \( C = -26 \)
- \( C = 28 \)
- \( C = -24 \)
- \( C = 16 \)
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The Correct Option is A
Solution and Explanation
We are given the line equation \( Ax + By + C = 0 \), which passes through the point \( (-10, 7) \) and is perpendicular to the line \( 11x - 8y - 16 = 0 \).
1. Step 1: Find the slope of the given line The slope of the line \( 11x - 8y - 16 = 0 \) can be found by rewriting the equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope: \[ 11x - 8y - 16 = 0 \quad \Rightarrow \quad -8y = -11x + 16 \quad \Rightarrow \quad y = \frac{11}{8}x - 2 \] Thus, the slope of this line is \( m_1 = \frac{11}{8} \).
2. Step 2: Find the slope of the perpendicular line The slope of the line perpendicular to this one is the negative reciprocal of \( \frac{11}{8} \), which is: \[ m_2 = -\frac{8}{11} \]
3. Step 3: Use the point-slope form of the line equation The line passes through the point \( (-10, 7) \), so we use the point-slope form to write the equation of the line: \[ y - 7 = -\frac{8}{11}(x + 10) \] Simplifying this equation: \[ y - 7 = -\frac{8}{11}x - \frac{80}{11} \] \[ y = -\frac{8}{11}x + \frac{7}{11} - \frac{80}{11} = -\frac{8}{11}x - \frac{73}{11} \]
4. Step 4: Convert the equation to standard form Multiply through by 11 to eliminate the denominator: \[ 11y = -8x - 73 \] Rearranging into the standard form \( Ax + By + C = 0 \): \[ 8x + 11y + 73 = 0 \] Thus, \( A = 8 \), \( B = 11 \), and \( C = -26 \).
1. Step 1: Find the slope of the given line The slope of the line \( 11x - 8y - 16 = 0 \) can be found by rewriting the equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope: \[ 11x - 8y - 16 = 0 \quad \Rightarrow \quad -8y = -11x + 16 \quad \Rightarrow \quad y = \frac{11}{8}x - 2 \] Thus, the slope of this line is \( m_1 = \frac{11}{8} \).
2. Step 2: Find the slope of the perpendicular line The slope of the line perpendicular to this one is the negative reciprocal of \( \frac{11}{8} \), which is: \[ m_2 = -\frac{8}{11} \]
3. Step 3: Use the point-slope form of the line equation The line passes through the point \( (-10, 7) \), so we use the point-slope form to write the equation of the line: \[ y - 7 = -\frac{8}{11}(x + 10) \] Simplifying this equation: \[ y - 7 = -\frac{8}{11}x - \frac{80}{11} \] \[ y = -\frac{8}{11}x + \frac{7}{11} - \frac{80}{11} = -\frac{8}{11}x - \frac{73}{11} \]
4. Step 4: Convert the equation to standard form Multiply through by 11 to eliminate the denominator: \[ 11y = -8x - 73 \] Rearranging into the standard form \( Ax + By + C = 0 \): \[ 8x + 11y + 73 = 0 \] Thus, \( A = 8 \), \( B = 11 \), and \( C = -26 \).
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