Question

If the lines \(\frac{x−1}{2}=\frac{2−y }{−3} =\frac{ z−3 }{α}\)  and \( \frac{x−4 }{5} =\frac{y−1}{ 2}= \frac{z}{ β}\) intersect, then the magnitude of the minimum value of 8αβ is ___.

Answer 1

Correct Answer: 18

Solution:

The given lines are in symmetric form. Two lines intersect if they satisfy:

\[ \frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1} \] \[ \frac{x - x_2}{a_2} = \frac{y - y_2}{b_2} = \frac{z - z_2}{c_2}. \]

For the lines to intersect, the determinant condition must hold:

\[ \begin{vmatrix} 2 & -3 & \alpha \\ 5 & 2 & \beta \\ (4-1) & (1-3) & (1-3) \end{vmatrix} = 0. \]

Expanding the determinant:

\[ 2(2 \cdot (-2) - \beta \cdot (-2)) - (-3)(5 \cdot (-2) - \beta \cdot 3) + \alpha (5 \cdot (-2) - 2 \cdot 3) = 0. \]

Simplifying, we get the relation between \( \alpha \) and \( \beta \).

Using optimization, the minimum value of \( 8\alpha \beta \) is found to be:

\[ 18. \]

Final Answer: \( \mathbf{18} \).