Question

The straight line \( ax + by + c = 0 \) passes through the point \( (-10,7) \). If the line is perpendicular to \( 11x - 7y = 13 \), then the value of \( c \) is equal to

• A. \(8 \)
• B. \(-7 \)
• C. \(13 \)
• D. \(-13 \)
• E. \(5 \)

Hint:

For perpendicular lines: \( m_1 m_2 = -1 \).

Answer 1

The Correct Option is B

Note: The provided source material indicates that this question was cancelled. However, we can solve it as follows.
Step 1: Understanding the Concept:
The problem requires finding the equation of a line given a point it passes through and a line it is perpendicular to. This involves finding the slope from the given line and using the condition for perpendicular lines.
Step 2: Key Formula or Approach:
1. Find the slope of the given line \(11x - 7y = 13\). Let this be \(m_1\). The slope of a line \(Ax + By + C = 0\) is \(-A/B\). 2. The slope of a line perpendicular to it, \(m_2\), is given by \(m_2 = -1/m_1\). 3. Use the point-slope form of a line, \(y - y_1 = m_2(x - x_1)\), to find the equation of the required line. 4. Convert the equation to the form \(ax + by + c = 0\) and identify the value of `c`.
Step 3: Detailed Explanation:
1. Find the slope of the given line. The line is \(11x - 7y = 13\). Its slope is \(m_1 = -\frac{\text{coefficient of x}}{\text{coefficient of y}} = -\frac{11}{-7} = \frac{11}{7}\). 2. Find the slope of the perpendicular line. The slope of our required line, \(m_2\), is the negative reciprocal of \(m_1\). \[ m_2 = -\frac{1}{m_1} = -\frac{1}{11/7} = -\frac{7}{11} \] 3. Find the equation of the required line. The line passes through the point \((x_1, y_1) = (-10, 7)\) and has a slope \(m_2 = -7/11\). Using the point-slope form: \[ y - 7 = -\frac{7}{11}(x - (-10)) \] \[ y - 7 = -\frac{7}{11}(x + 10) \] Multiply both sides by 11 to eliminate the fraction: \[ 11(y - 7) = -7(x + 10) \] \[ 11y - 77 = -7x - 70 \] Rearrange the equation into the form \(ax + by + c = 0\): \[ 7x + 11y - 77 + 70 = 0 \] \[ 7x + 11y - 7 = 0 \] 4. Identify the value of c. Comparing this equation with \(ax + by + c = 0\), we see that \(c = -7\).
Step 4: Final Answer:
The calculated value of c is -7.