Coordinate Geometry
Locus of intersection of two variable lines
MMTS_Full_Test_17
Grade 12
Question:
Given lines $\dfrac{x}{a}+\dfrac{y}{b}=1$ and $ax+by=1$ are two variable lines, $a$ and $b$ being parameters connected by $a^2+b^2=ab$. The locus of the point of intersection is
(A) $x^2+y^2+xy-1=0$
(B) $x^2+y^2-xy+1=0$
(C) $x^2+y^2+xy+1=0$
(D) $x^2+y^2-xy-1=0$
Step-by-Step Solution
Key Concept: Let $(h,k)$ be the intersection. Then $h/a+k/b=1$ and $ah+bk=1$. Multiply: $h^2+k^2+hk(b/a+a/b)=1$. Use $a^2+b^2=ab \Rightarrow a/b+b/a=1$.
Intersection: $h^2+k^2+hk(a/b+b/a)=1$. Since $a/b+b/a=1$: locus $x^2+y^2+xy-1=0$.
Correct Answer: (A) $x^2+y^2+xy-1=0$