Key Concepts and Formulas

• Angle Between Two Lines: If two lines have slopes $m_1$ and $m_2$, and $\theta$ is the angle between them, then $\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$.
• Point-Slope Form of a Line: The equation of a line passing through $(x_1, y_1)$ with slope $m$ is $y - y_1 = m(x - x_1)$.
• Slope-Intercept Form of a Line: The equation of a line with slope $m$ and y-intercept $c$ is $y = mx + c$.

Step-by-Step Solution

Step 1: Identify the given information and find the slope of the given line.

We are given the point $(1, 3)$ and the angle $\theta = \tan^{-1}(\sqrt{2})$, which means $\tan\theta = \sqrt{2}$. The equation of the other line is $y + 1 = 3\sqrt{2}x$. We need to find its slope. Rewriting the equation in slope-intercept form, we get $y = 3\sqrt{2}x - 1$. Therefore, the slope of the given line is $m_1 = 3\sqrt{2}$.

Step 2: Apply the angle between two lines formula.

Let $m_2 = m$ be the slope of the required line. We use the formula for the tangent of the angle between two lines: $\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$ Substituting the given values, we have: $\sqrt{2} = \left| \frac{3\sqrt{2} - m}{1 + 3\sqrt{2}m} \right|$

Step 3: Remove the absolute value and solve for m.

Since we have an absolute value, we need to consider two cases:

Case 1: $\frac{3\sqrt{2} - m}{1 + 3\sqrt{2}m} = \sqrt{2}$ Multiplying both sides by $(1 + 3\sqrt{2}m)$, we get: $3\sqrt{2} - m = \sqrt{2}(1 + 3\sqrt{2}m)$ $3\sqrt{2} - m = \sqrt{2} + 6m$ $2\sqrt{2} = 7m$ $m = \frac{2\sqrt{2}}{7}$

Case 2: $\frac{3\sqrt{2} - m}{1 + 3\sqrt{2}m} = -\sqrt{2}$ Multiplying both sides by $(1 + 3\sqrt{2}m)$, we get: $3\sqrt{2} - m = -\sqrt{2}(1 + 3\sqrt{2}m)$ $3\sqrt{2} - m = -\sqrt{2} - 6m$ $4\sqrt{2} = -5m$ $5m = -4\sqrt{2}$ $m = -\frac{4\sqrt{2}}{5}$

Step 4: Find the equations of the lines using the point-slope form.

We have two possible slopes and the point $(1, 3)$. We use the point-slope form $y - y_1 = m(x - x_1)$.

Case 1: $m = \frac{2\sqrt{2}}{7}$ $y - 3 = \frac{2\sqrt{2}}{7}(x - 1)$ $7(y - 3) = 2\sqrt{2}(x - 1)$ $7y - 21 = 2\sqrt{2}x - 2\sqrt{2}$ $2\sqrt{2}x - 7y + 21 - 2\sqrt{2} = 0$ This equation is $2\sqrt{2}x - 7y + (21 - 2\sqrt{2}) = 0$, which doesn't match any of the options.

Case 2: $m = -\frac{4\sqrt{2}}{5}$ $y - 3 = -\frac{4\sqrt{2}}{5}(x - 1)$ $5(y - 3) = -4\sqrt{2}(x - 1)$ $5y - 15 = -4\sqrt{2}x + 4\sqrt{2}$ $4\sqrt{2}x + 5y - 15 - 4\sqrt{2} = 0$ $4\sqrt{2}x + 5y - (15 + 4\sqrt{2}) = 0$ This matches option (A).

Step 5: Select the matching option.

The equation $4\sqrt{2}x + 5y - (15 + 4\sqrt{2}) = 0$ matches option (A).

Common Mistakes & Tips

• Forgetting the absolute value: Always remember to consider both positive and negative cases when removing the absolute value.
• Algebraic errors: Be careful with algebraic manipulations when solving for $m$.
• Not checking both slopes: Remember to find the equation of the line for both possible slopes.

Summary

We used the formula for the angle between two lines to find two possible slopes for the required line. Then, we used the point-slope form to find the equations of the lines. Only one of the equations matched one of the given options. Therefore, we selected that option. The equation of one of the straight lines is $4\sqrt{2}x + 5y - (15 + 4\sqrt{2}) = 0$.

Final Answer

The final answer is $\boxed{4\sqrt 2 x + 5y - \left( {15 + 4\sqrt 2 } \right) = 0}$, which corresponds to option (A).