Question

The equation of one of the straight lines which passes through the point (1, 3) and makes an angle tan⁻¹(√2) with the straight line, y + 1 = 3√2 x is :

• A. 4√2 x + 5y - (15 + 4√2) = 0
• B. 4√2 x - 5y - (5 + 4√2) = 0
• C. 5√2 x + 4y - (15 + 4√2) = 0
• D. 4√2 x + 5y - 4√2 = 0

Hint:

Remember to solve for both cases of the absolute value when finding slopes of lines at an angle.

Answer 1

The Correct Option is A

To solve this problem, we need to determine the equation of a straight line that passes through the point \((1, 3)\) and makes an angle of \(\tan^{-1}(\sqrt{2})\) with the given line \(y + 1 = 3\sqrt{2} x\).

1. First, identify the slope of the given line. The equation \(y + 1 = 3\sqrt{2} x\) can be rewritten in the slope-intercept form \(y = mx + c\), where \(m\) is the slope.

   The equation becomes:

   y = 3\sqrt{2} x - 1

   Thus, the slope m_1 = 3\sqrt{2}.

2. The required line makes an angle of \(\theta = \tan^{-1}(\sqrt{2})\) with this line. Therefore, if m_2 is the slope of the required line, then:

   \frac{m_2 - m_1}{1 + m_2 \cdot m_1} = \sqrt{2}

3. Substitute m_1 = 3\sqrt{2} and solve for \(m_2\):

   \frac{m_2 - 3\sqrt{2}}{1 + 3\sqrt{2} \cdot m_2} = \sqrt{2}

   Cross-multiply to get:

   m_2 - 3\sqrt{2} = \sqrt{2}(1 + 3\sqrt{2} \cdot m_2)

   Expanding and simplifying:

   m_2 - 3\sqrt{2} = \sqrt{2} + 6m_2 m_2 - 6m_2 = \sqrt{2} + 3\sqrt{2} -5m_2 = 4\sqrt{2} m_2 = -\frac{4\sqrt{2}}{5}

4. Now, we have the slope of the required line as m_2 = -\frac{4\sqrt{2}}{5} and the line passes through the point (1, 3).

   Therefore, using the point-slope form:

   y - 3 = -\frac{4\sqrt{2}}{5}(x - 1)

   Expanding this equation gives:

   y - 3 = -\frac{4\sqrt{2}}{5}x + \frac{4\sqrt{2}}{5}

   Multiplying everything by 5 to eliminate fractions:

   5y - 15 = -4\sqrt{2}x + 4\sqrt{2}

   Rearranging terms, we get:

   4\sqrt{2}x + 5y - (15 + 4\sqrt{2}) = 0

5. Thus, the equation of the required line is:

   4\sqrt{2}x + 5y - (15 + 4\sqrt{2}) = 0

   This matches the given option:

   4√2 x + 5y - (15 + 4√2) = 0

This confirms that the option 4√2 x + 5y - (15 + 4√2) = 0 is the correct answer.