Question

If the line passing through the point \((4, -3)\) and having negative slope makes an angle of \(45^\circ\) with the line joining the points \((1,1), (2,3)\) then the sum of intercepts of that line is

• A. 3
• B. 1
• C. 12
• D. \(\frac{26}{3}\)

Hint:

When finding the slope from the angle between two lines, remember there are usually two possible slopes (\(m\) and \(m'\)). Always check additional conditions given in the problem (like "negative slope") to select the correct one.

Answer 1

The Correct Option is C

Step 1: Find slope of the given line segment:
Slope \(m_1\) of the line joining \((1,1)\) and \((2,3)\) is: \[ m_1 = \frac{3 - 1}{2 - 1} = \frac{2}{1} = 2 \]
Step 2: Use the angle formula to find the slope of the required line:
Let the slope of the required line be \(m\). The angle between the lines is \(45^\circ\). \[ \tan 45^\circ = \left| \frac{m - m_1}{1 + m m_1} \right| \] \[ 1 = \left| \frac{m - 2}{1 + 2m} \right| \] This gives two cases: Case 1: \(\frac{m - 2}{1 + 2m} = 1 \implies m - 2 = 1 + 2m \implies m = -3\). Case 2: \(\frac{m - 2}{1 + 2m} = -1 \implies m - 2 = -1 - 2m \implies 3m = 1 \implies m = 1/3\). The problem states the line has a negative slope, so we choose \(m = -3\).
Step 3: Equation of the line:
The line passes through \((4, -3)\) with slope \(m = -3\). \[ y - (-3) = -3(x - 4) \] \[ y + 3 = -3x + 12 \] \[ 3x + y - 9 = 0 \]
Step 4: Find Intercepts:
x-intercept (set \(y=0\)): \(3x = 9 \implies x = 3\). y-intercept (set \(x=0\)): \(y = 9\).
Step 5: Sum of intercepts:
Sum = \(3 + 9 = 12\).