Key Concepts and Formulas

• Parallel Lines: If two lines are perpendicular to the same line, they are parallel to each other.
• Slope of a Line: The slope of a line given by the equation $Ax + By + C = 0$ is $m = -\frac{A}{B}$, provided $B \neq 0$.
• Condition for Parallel Lines (Slopes): Two lines are parallel if and only if their slopes are equal (i.e., $m_1 = m_2$).

Step-by-Step Solution

Step 1: Identify the coefficients of the given lines. We have two lines: Line 1 ($L_1$): $p(p^2 + 1)x - y + q = 0$ Line 2 ($L_2$): $(p^2 + 1)^2x + (p^2 + 1)y + 2q = 0$

We extract the coefficients for each line, noting the standard form $Ax + By + C = 0$:

For Line 1: $A_1 = p(p^2 + 1)$ $B_1 = -1$ $C_1 = q$

For Line 2: $A_2 = (p^2 + 1)^2$ $B_2 = (p^2 + 1)$ $C_2 = 2q$

Step 2: Calculate the slope of the first line ($L_1$). Using the slope formula $m = -\frac{A}{B}$: $m_1 = -\frac{p(p^2 + 1)}{-1}$ Simplifying, we get: $m_1 = p(p^2 + 1)$

Step 3: Calculate the slope of the second line ($L_2$). Using the slope formula $m = -\frac{A}{B}$: $m_2 = -\frac{(p^2 + 1)^2}{p^2 + 1}$ Since $p^2 \ge 0$ for all real $p$, we have $p^2 + 1 \ge 1 > 0$. Thus, $p^2 + 1 \neq 0$, and we can simplify: $m_2 = -(p^2 + 1)$

Step 4: Apply the condition for parallel lines. Since the lines are perpendicular to a common line, they must be parallel to each other. Therefore, their slopes must be equal: $m_1 = m_2$ Substituting the expressions for $m_1$ and $m_2$, we have: $p(p^2 + 1) = -(p^2 + 1)$

Step 5: Solve the equation for $p$. We need to find the value(s) of $p$ that satisfy this equation. $p(p^2 + 1) = -(p^2 + 1)$ Move all terms to one side: $p(p^2 + 1) + (p^2 + 1) = 0$ Factor out the common factor $(p^2 + 1)$: $(p^2 + 1)(p + 1) = 0$ For the product to be zero, at least one factor must be zero.

Case 1: $p^2 + 1 = 0$ This implies $p^2 = -1$. Since $p$ is real, $p^2$ must be non-negative. Therefore, there are no real solutions for $p$ in this case.

Case 2: $p + 1 = 0$ This implies $p = -1$.

Thus, the only real value of $p$ that satisfies the condition is $p = -1$.

Step 6: Determine the number of values for $p$. We found exactly one real value for $p$, which is $p = -1$.

Common Mistakes & Tips

• Understanding the Geometric Interpretation: Recognizing that lines perpendicular to a common line are parallel is crucial.
• Dividing by Zero: Be careful when dividing by expressions involving variables. Ensure the expression is not zero. In this case, $p^2 + 1$ is always positive for real $p$. Factoring is generally safer than dividing.
• Real Solutions: Remember to consider only real solutions unless otherwise specified.

Summary

The problem tests the understanding of the relationship between perpendicular and parallel lines. By equating the slopes of the two lines (since they are parallel), we found the condition on $p$. Solving the resulting equation, we found that there is exactly one real value of $p$ for which the condition holds true, namely $p=-1$.

Final Answer The final answer is \boxed{exactly one values of $p$}, which corresponds to option (A).