Straight Lines
Locus — Variable Point with Fixed Triangle Area
nta_pyq_2024_apr
Grade 11
Question:
Let $A(-1,1)$ and $B(2,3)$ be two points and $P$ be a variable point above the line $AB$ such that the area of $\triangle PAB$ is 10. If the locus of $P$ is $ax+by=15$, then $5a+2b$ is:
6
$-\dfrac{6}{5}$
4
$-\dfrac{12}{5}$
Step-by-Step Solution
Key Concept: Area of $\triangle PAB=\frac{1}{2}\left|\det\begin{pmatrix}h-(-1)&k-1\\2-(-1)&3-1\end{pmatrix}\right|=10$. This gives $|2(h+1)-3(k-1)|=20$, i.e. $-2h+3k=25$ (taking the branch above line $AB$).
$a=-6/5$, $b=9/5$. $5a+2b=-12/5$.
Correct Answer: 4