The coordinates of the point of intersection of the lines $\frac{x-3}{1} = \frac{y-5}{2} = \frac{z-1}{-1}$ and $\frac{x-4}{2} = \frac{y-2}{-1} = \frac{z-4}{2}$ are...

Question

The angle between the lines, whose direction cosines $ l, m, n $ satisfy the equations: \[ l + m + n = 0 \quad \text{and} \quad 2l^2 + 2m^2 - n^2 = 0, \] is:

Answer 1

Step 1: Test option A in both lines:
For line 1: $\frac{2 - 3}{1} = -1$, $\frac{3 - 5}{2} = -1$, $\frac{2 - 1}{-1} = -1$. All equal.

Step 2: Line 2:
$\frac{2 - 4}{2} = -1$, $\frac{3 - 2}{-1} = -1$, $\frac{2 - 4}{2} = -1$. All equal.
So $(2, 3, 2)$ lies on both lines. The other options give unequal ratios.

Final Answer:
The intersection point is $(2, 3, 2)$, option (A). \[ \boxed{(2,3,2)} \]

The coordinates of the point of intersection of the lines \(\frac{x-3}{1} = \frac{y-5}{2} = \frac{z-1}{-1}\) and \(\frac{x-4}{2} = \frac{y-2}{-1} = \frac{z-4}{2}\) are...

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The Correct Option is A

Solution and Explanation

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