Question

Find the distance of the point $(-1, -5, -10)$ from the point of intersection of the lines \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}, \quad \frac{x - 4}{5} = \frac{y - 1}{2} = z. \]

Hint:

To find the distance from a point to a line or between two points, use the distance formula, which is derived from the Pythagorean theorem.

Answer 1

To solve this, first find the intersection of the two lines. Parametrize both lines. For the first line, let: \[ \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} = t. \] Thus, \[ x = 2t + 1, \quad y = 3t + 2, \quad z = 4t + 3. \] For the second line, let: \[ \frac{x - 4}{5} = \frac{y - 1}{2} = z = s. \] Thus, \[ x = 5s + 4, \quad y = 2s + 1, \quad z = s. \] Equating the expressions for $x$, $y$, and $z$ from both lines, we solve for $s$ and $t$. Once the intersection point is found, calculate the distance from $(-1, -5, -10)$ to the intersection point using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}. \]