Question:

Step 1: Understand the relationship between direction cosines and direction ratios. If (l, m, n) are the direction cosines of a line, they satisfy: l^2 + m^2 + n^2 = 1 Step 2: Recognize what the equation \(\frac{x}{a} = \frac{y}{b} = \frac{z}{c}\) represents. This is the symmetric form of a line through the origin, where (a, b, c) are the direction ratios (or direction numbers), not necessarily direction cosines. Step 3: Determine the relationship between direction cosines (l, m, n) and direction ratios (a, b, c). Direction ratios are proportional to direction cosines. If (l, m, n) are direction cosines, then (a, b, c) = k(l, m, n) for some non-zero constant k. Step 4: Find the constraint on direction cosines. Since l^2 + m^2 + n^2 = 1, we have: • l^2 = 1 - m^2 - n^2 • m^2 = 1 - l^2 - n^2 • n^2 = 1 - l^2 - m^2 Step 5: Analyze option (a): (\(\sqrt{1 - l^2}, \sqrt{1 - m^2}, \sqrt{1 - n^2}\)) From l^2 + m^2 + n^2 = 1: • 1 - l^2 = m^2 + n^2 • 1 - m^2 = l^2 + n^2 • 1 - n^2 = l^2 + m^2 So the triad becomes: (\(\sqrt{m^2 + n^2}, \sqrt{l^2 + n^2}, \sqrt{l^2 + m^2}\)) This represents direction ratios proportional to the direction cosines (l, m, n) when properly normalized, and satisfies the relationship for a line with direction cosines (l, m, n). Step 6: Verify this is the standard form. The symmetric form \(\frac{x}{a} = \frac{y}{b} = \frac{z}{c}\) with these direction ratios correctly represents a line whose direction cosines are (l, m, n). ∴ Answer: A