3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade None
Question:
The line $\frac{x + 6}{5} = \frac{y + 10}{3} = \frac{z + 14}{8}$ is the hypotenuse of an isosceles right angled triangle whose opposite vertex is $(7, 2, 4)$. The equations of the remaining sides are:
$\frac{x - 7}{5} = \frac{y - 2}{2} = \frac{z - 4}{6}$
$\frac{x - 7}{3} = \frac{y - 2}{6} = \frac{z - 4}{2}$
$\frac{x - 7}{-2} = \frac{y - 2}{-2} = \frac{z - 4}{6}$
$\frac{x - 7}{2} = \frac{y - 2}{-3} = \frac{z - 4}{6}$
Step-by-Step Solution
Key Concept: The angle between two lines is found using the dot product formula with their direction vectors, then solve for the parameter.
Point $B$ has coordinates $(-6 + 5\lambda, -10 + 3\lambda, -14 + 8\lambda)$, giving direction ratio $AB$ as $5\lambda - 13, 3\lambda - 12, 8\lambda - 18$. The angle between $AB$ and $BC$ is $45°$, which occurs when $\lambda = 2$ and $\lambda = 3$. Using the angle condition, we can verify the specific value of $\lambda$ that satisfies the geometry.
Correct Answer: I need to find the equations of the two remaining sides of an isosceles right-angled triangle where the hypotenuse is given and the opposite vertex is (7, 2, 4).
**Key Setup:**
- Hypotenuse: $\frac{x + 6}{5} = \frac{y + 10}{3}