CBSE 2025, Set 3

Question

**Case Study - 2**
An engineer is designing a new metro rail network in a city.

![Diagram Metro Rail](https://www.edvaya.com/images/cbse/2025/mathematics/set3/q37-metro-rail.webp)

Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by
$l_1: \frac{x-2}{3} = \frac{y+1}{-2} = \frac{z-3}{4}$ while the track for Line B is represented by
$l_2: \frac{x-1}{2} = \frac{y-3}{1} = \frac{z+2}{-3}$.

Based on the above information, answer the following question:
(ii) Solar panels are to be installed on the rooftop of the metro stations. Determine the equation of the line representing the placement of solar panels on the rooftop of Line A's stations, given that panels are to be positioned parallel to Line A's track ($l_1$) and pass through the point $(1, -2, -3)$.

Edvaya · Accepted Answer

The correct answer is Subjective. **Step 1:** Identify the given point and the required direction vector. The line representing the solar panels passes through the point $(x_1, y_1, z_1) = (1, -2, -3)$. Since this new line is strictly parallel to Line A ($l_1$), its direction ratios must be proportional (or equal) to the direction ratios of Line A. From the equation of Line A, its direction ratios are $(3, -2, 4)$. Thus, the direction vector for the solar panels is $\vec{b} = 3\hat{i} - 2\hat{j} + 4\hat{k}$. **Step 2:** Formulate the cartesian equation of the line. The standard equation of a line passing through a point $(x_1, y_1, z_1)$ with direction ratios $(a, b, c)$ is: $$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$$ Substituting the known values: $$\frac{x - 1}{3} = \frac{y - (-2)}{-2} = \frac{z - (-3)}{4}$$ $$\frac{x - 1}{3} = \frac{y + 2}{-2} = \frac{z + 3}{4}$$ This is the required equation of the line.

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