Trigonometry & Inverse Trigonometry
Orthocentre via Euler Line — 3D Coordinate Geometry
nta_pyq_2023_apr
Grade 11
Question:
Let the plane $x+3y-2z+6=0$ meet the coordinate axes at $A$, $B$, $C$. If the orthocentre of $\triangle ABC$ is $\left(\alpha,\beta,\dfrac{6}{7}\right)$, then $98(\alpha+\beta)^2$ is equal to __________.
Step-by-Step Solution
Key Concept: Intercepts: $A=(-6,0,0)$, $B=(0,-2,0)$, $C=(0,0,3)$. Find circumcentre via perpendicular bisectors of sides. Use Euler line (centroid divides $O$ to $H$ in $1:2$) to find orthocentre.
$\alpha+\beta=-\frac{12}{7}$. $98(\alpha+\beta)^2=288$.
Correct Answer: 288