This problem asks us to find the condition for two lines, given in a non-symmetric form, to be perpendicular in three-dimensional space. The standard approach involves converting the line equations into their symmetric form to easily identify their direction ratios (DRs), and then applying the perpendicularity condition.

1. Key Concepts and Formulas

• Standard Symmetric Form of a Line: A line passing through a point $(x_1, y_1, z_1)$ with direction ratios $(l, m, n)$ can be written as: $\frac{x-x_1}{l} = \frac{y-y_1}{m} = \frac{z-z_1}{n}$
• Direction Ratios (DRs) from Non-Symmetric Form: If a line is given by two equations (e.g., as the intersection of two planes), one method to find its DRs is to express $x, y, z$ in terms of a common parameter. For example, if $x = py + q$ and $z = ry + s$, then by setting $y = \lambda$, we get $x = p\lambda + q$, $y = 1\lambda + 0$, $z = r\lambda + s$. The direction ratios are then $(p, 1, r)$.
• Perpendicularity Condition for Two Lines: Two lines with direction ratios $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ are perpendicular if and only if the dot product of their direction vectors is zero: $l_1l_2 + m_1m_2 + n_1n_2 = 0$

2. Step-by-Step Solution

Our goal is to transform the given equations for each line into the standard symmetric form to extract their direction ratios.

Step 1: Determine Direction Ratios for the First Line The first line is given by the equations:

1. $x = ay + b$
2. $z = cy + d$

• Reasoning: These equations express $x$ and $z$ in terms of $y$. We can treat $y$ as a parameter (say, $\lambda$). From equation (1), we can write $\frac{x-b}{a} = y$. From equation (2), we can write $\frac{z-d}{c} = y$. Also, $y$ can be written as $\frac{y-0}{1}$.
• Mathematical Derivation: Equating these expressions, we get the symmetric form of the first line: $\frac{x-b}{a} = \frac{y-0}{1} = \frac{z-d}{c}$
• Identifying DRs: By comparing this to the standard form $\frac{x-x_1}{l_1} = \frac{y-y_1}{m_1} = \frac{z-z_1}{n_1}$, the direction ratios for the first line are $(l_1, m_1, n_1) = (a, 1, c)$.

Step 2: Determine Direction Ratios for the Second Line The second line is given by the equations: 3. $x = a'z + b'$ 4. $y = c'z + d'$

• Reasoning: These equations express $x$ and $y$ in terms of $z$. We can treat $z$ as a parameter (say, $\mu$). From equation (3), we can write $\frac{x-b'}{a'} = z$. From equation (4), we can write $\frac{y-d'}{c'} = z$. Also, $z$ can be written as $\frac{z-0}{1}$.
• Mathematical Derivation: Equating these expressions, we get the symmetric form of the second line: $\frac{x-b'}{a'} = \frac{y-d'}{c'} = \frac{z-0}{1}$
• Identifying DRs: By comparing this to the standard form $\frac{x-x_2}{l_2} = \frac{y-y_2}{m_2} = \frac{z-z_2}{n_2}$, the direction ratios for the second line are $(l_2, m_2, n_2) = (a', c', 1)$.

Step 3: Apply the Perpendicularity Condition Now that we have the direction ratios for both lines, we apply the condition for perpendicularity: $l_1l_2 + m_1m_2 + n_1n_2 = 0$

• Mathematical Derivation: Substitute the identified DRs: $(a)(a') + (1)(c') + (c)(1) = 0$ $aa' + c' + c = 0$

This is the condition for the two given lines to be perpendicular.

3. Common Mistakes & Tips

• Converting to Symmetric Form: The most common mistake is incorrectly identifying the direction ratios. Always ensure the symmetric form is in the format $\frac{x-x_1}{l} = \frac{y-y_1}{m} = \frac{z-z_1}{n}$. If a variable is implicitly the parameter (e.g., $y$ in $x=ay+b, z=cy+d$), its coefficient in the denominator of its own term will be 1 (e.g., $\frac{y-0}{1}$).
• Distinguishing Direction Ratios from Point Coordinates: Remember that the constants $b, d, b', d'$ (or their negatives) are related to the coordinates of a point on the line, not the direction ratios. The direction ratios are the coefficients of the parameter in the parametric form or the denominators in the symmetric form.
• Handling Zero Denominators: If a direction ratio is zero (e.g., if $a=0$, meaning $x=b$), the symmetric form becomes $\frac{x-b}{0} = \frac{y}{1} = \frac{z-d}{c}$. A zero in the denominator implies that the numerator must be zero for any point on the line, and that the direction vector component is zero. The DRs $(a, 1, c)$ are still valid even if $a$ or $c$ is zero.

4. Summary

To find the condition for perpendicularity of lines given in non-symmetric form, first convert each line's equations into the standard symmetric form. From this, extract the direction ratios $(l, m, n)$ for each line. Finally, apply the perpendicularity condition $l_1l_2 + m_1m_2 + n_1n_2 = 0$. Following these steps for the given lines, the condition for perpendicularity is $aa' + c + c' = 0$.

The final answer is $\boxed{aa' + c + c' = 0}$, which corresponds to option (D).