Key Concepts and Formulas

• Slope of a line: The slope of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• Angle between two lines: If $\theta$ is the angle between two lines with slopes $m_1$ and $m_2$, then $\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$. This formula holds when $1 + m_1m_2 \neq 0$.
• Absolute value property: If $|x| = a$, where $a > 0$, then $x = a$ or $x = -a$.

Step-by-Step Solution

Step 1: Define the points and identify the required slopes. We are given points $P(5, 2)$ and $Q(2, a)$. We need to find the angle between the lines $OP$ and $OQ$, where $O$ is the origin $(0, 0)$. Let $m_1$ be the slope of $OP$ and $m_2$ be the slope of $OQ$. Explanation: This step sets up the problem by defining the points and identifying the key slopes we need to calculate.

Step 2: Calculate the slope of line $OP$. Using the slope formula with points $O(0, 0)$ and $P(5, 2)$: $m_1 = \frac{2 - 0}{5 - 0} = \frac{2}{5}$ Explanation: We apply the slope formula to find the slope of the line segment connecting the origin and point P.

Step 3: Calculate the slope of line $OQ$. Using the slope formula with points $O(0, 0)$ and $Q(2, a)$: $m_2 = \frac{a - 0}{2 - 0} = \frac{a}{2}$ Explanation: We apply the slope formula to find the slope of the line segment connecting the origin and point Q.

Step 4: Apply the angle formula. We are given that the angle between the lines is $\frac{\pi}{4}$, so $\tan\left(\frac{\pi}{4}\right) = 1$. Plugging the slopes $m_1$ and $m_2$ into the angle formula: $1 = \left| \frac{\frac{a}{2} - \frac{2}{5}}{1 + \frac{a}{2} \cdot \frac{2}{5}} \right|$ Explanation: We substitute the known slopes and angle into the formula for the tangent of the angle between two lines.

Step 5: Simplify the expression inside the absolute value. $1 = \left| \frac{\frac{5a - 4}{10}}{1 + \frac{a}{5}} \right| = \left| \frac{\frac{5a - 4}{10}}{\frac{5 + a}{5}} \right| = \left| \frac{5a - 4}{10} \cdot \frac{5}{5 + a} \right| = \left| \frac{5a - 4}{2(5 + a)} \right| = \left| \frac{5a - 4}{10 + 2a} \right|$ Explanation: We simplify the complex fraction within the absolute value to make it easier to solve.

Step 6: Solve the absolute value equation. Since $|x| = 1$, we have two cases: $x = 1$ or $x = -1$.

Case 1: $\frac{5a - 4}{10 + 2a} = 1$ $5a - 4 = 10 + 2a$ $3a = 14$ $a = \frac{14}{3}$

Case 2: $\frac{5a - 4}{10 + 2a} = -1$ $5a - 4 = -10 - 2a$ $7a = -6$ $a = -\frac{6}{7}$ Explanation: We consider both possible cases arising from the absolute value and solve for $a$ in each case.

Step 7: Calculate the product of the possible values of $a$ and its absolute value. The two possible values for $a$ are $\frac{14}{3}$ and $-\frac{6}{7}$. Their product is: $P = \frac{14}{3} \cdot \left(-\frac{6}{7}\right) = -\frac{14 \cdot 6}{3 \cdot 7} = -\frac{2 \cdot 7 \cdot 2 \cdot 3}{3 \cdot 7} = -4$ The absolute value of the product is: $|P| = |-4| = 4$ Explanation: We calculate the product of the two possible values of $a$ and then find the absolute value of that product.

Common Mistakes & Tips

• Forgetting the negative case in absolute value equations: Remember to consider both positive and negative possibilities when solving an absolute value equation.
• Arithmetic errors: Be careful with fractions and simplification steps to avoid arithmetic mistakes.
• Checking for extraneous solutions: While not strictly necessary here, always check if your solutions make the denominator zero in any of the intermediate steps.

Summary

We used the formula for the angle between two lines to set up an equation involving the unknown $a$. By considering both positive and negative cases of the absolute value, we found two possible values of $a$. The absolute value of the product of these values is 4.

The final answer is $\boxed{4}$, which corresponds to option (A).