<p>If \(l + 3m + 5n = 0\) and \(5lm - 2mn + 6nl = 0\), then the directions of the two lines satisfying these conditions give \(m =\):</p>
Step-by-Step Solution
Key Concept: The two given equations represent conditions on direction cosines (l, m, n) of two lines. From the linear constraint, express one variable in terms of others, then substitute into the quadratic constraint to find the ratio m:l or m:n for each line.
Step 1: From the linear constraint: l + 3m + 5n = 0, express n = -(l + 3m)/5 Step 2: Substitute into the quadratic equation 5lm - 2mn + 6nl = 0: 5lm - 2m(-(l + 3m)/5) + 6l(-(l + 3m)/5) = 0 5lm + 2m(l + 3m)/5 - 6l(l + 3m)/5 = 0 Step 3: Multiply by 5: 25lm + 2m(l + 3m) - 6l(l + 3m) = 0 25lm + 2lm + 6m^2 - 6l^2 - 18lm = 0 9lm + 6m^2 - 6l^2 = 0 Divide by l^2: 9(m/l) + 6(m/l)^2 - 6 = 0 Step 4: Let t = m/l: 6t^2 + 9t - 6 = 0 or 2t^2 + 3t - 2 = 0 (2t - 1)(t + 2) = 0 Step 5: Therefore t = 1/2 or t = -2, giving m = l/2 or m = -2l ∴ Answer: A
Correct Answer: A