3D Geometry
Section formula and distance
nta_pyq_2023_jan
Grade None
Question:
The plane $2x - y + z = 4$ intersects the line segment joining the points $A(a, -2, 4)$ and $B(2, b, -3)$ at the point C in the ratio $2:1$ and the distance of the point C from the origin is $\sqrt{5}$. If $ab < 0$ and P is the point $(a-b, b, 2b-a)$ then $CP^2$ is equal to:
\frac{17}{3}
\frac{16}{3}
\frac{73}{3}
\frac{97}{3}
Step-by-Step Solution
Key Concept: Find C using section formula (2:1), apply plane condition, use $|OC|=\sqrt{5}$, then constraint $ab<0$ to determine $a,b$.
$(a,b)=(1,-1)$, $C=(5/3,-4/3,-2/3)$, $P=(2,-1,-3)$. $CP^2 = (2-5/3)^2+(-1+4/3)^2+(-3+2/3)^2 = 1/9+1/9+49/9 = 51/9 = 17/3$. Answer: (1)
Correct Answer: $\frac{17}{3}$