Question

If the line y = mx + c is perpendicular to y=1+x and passes through the point (1, 2), then the value of c is equal to

• A. 1
• B. -1
• C. -3
• D. 3
• E. 0

Answer 1

The Correct Option is D

The equation of the line is given by \( y = mx + c \), where \( m \) is the slope of the line.
The equation \( y = 1 + x \) is in the form \( y = mx + c \), so the slope of the given line is \( m = 1 \).
Since the two lines are perpendicular, the product of their slopes must be \( -1 \).
Let the slope of the line \( y = mx + c \) be \( m_1 \). Then, \[ m_1 \times 1 = -1 \] Thus, \( m_1 = -1 \). Now the equation of the line becomes: \[ y = -x + c \] We are given that the line passes through the point \( (1, 2) \). Substituting \( x = 1 \) and \( y = 2 \) into the equation \( y = -x + c \): \[ 2 = -(1) + c \] \[ 2 = -1 + c \] \[ c = 3 \]

The correct option is (D) : \(3\)

Answer 2

We are given the line \(y = mx + c\) and the line \(y = 1 + x\), which can be written as \(y = x + 1\).

The slope of the line \(y = x + 1\) is 1. Since the line \(y = mx + c\) is perpendicular to \(y = x + 1\), the product of their slopes must be -1.

Therefore, \(m \cdot 1 = -1\), which means \(m = -1\).

So, the equation of the line is \(y = -x + c\).

The line passes through the point (1, 2). Substituting x = 1 and y = 2 into the equation, we get:

\(2 = -1 + c\)

Solving for \(c\), we have:

\(c = 2 + 1 = 3\)

Therefore, the value of \(c\) is 3.