3D Geometry
Line Through Point Intersecting Two Lines
nta_pyq_2024_jan
Grade 12
Question:
Let a line passing through the point $(-1,2,3)$ intersect the lines $L_1:\dfrac{x-1}{3}=\dfrac{y-2}{2}=\dfrac{z+1}{-2}$ at $M(\alpha,\beta,\gamma)$ and $L_2:\dfrac{x+2}{-3}=\dfrac{y-2}{-2}=\dfrac{z-1}{4}$ at $N(a,b,c)$. Then the value of $\dfrac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$ equals
Step-by-Step Solution
Key Concept: A line through $(-1,2,3)$ intersects both $L_1$ and $L_2$. Parametrize $M$ on $L_1$ and $N$ on $L_2$. For $(-1,2,3)$, $M$, $N$ to be collinear, apply condition.
$\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}=196$.
Correct Answer: 196