Straight Lines
Orthocentre and Circumcentre — Combined Expression
nta_pyq_2024_apr
Grade 11
Question:
Two vertices of a triangle $ABC$ are $A(3,-1)$ and $B(-2,3)$, and its orthocentre is $P(1,1)$. If the coordinates of the point $C$ are $(\alpha,\beta)$ and the centre of the circle circumscribing the triangle $PAB$ is $(h,k)$, then the value of $(\alpha+\beta)+2(h+k)$ equals
Step-by-Step Solution
Key Concept: Find $C$: $CP\perp AB$ and $BP\perp AC$. Slope $AB=\frac{3-(-1)}{-2-3}=-4/5$, so $CP$ has slope $5/4$. Slope $AP=\frac{1-(-1)}{1-3}=-1$, so $BC$ has slope $1$.
$C=(21,26)$. Circumcentre of $\triangle PAB$: $(h,k)=(-19/2,-23/2)$. $(\alpha+\beta)+2(h+k)=47-42=5$.
Correct Answer: 1