Question

# $$AB$$ is a variable line sliding between the coordinate axes in such a way that $$A$$ lies on x-axis and $$B$$ lies on y-axis. If $$P$$ is a variable point on $$AB$$ such that $$PA = b, PB = a$$ and $$AB = a + b$$, find the equation of the locus of $$P$$.

Solution

## $$A(l,0),B(0,h)$$$$\therefore$$ By section formula$$P(x,y)=(\cfrac { al }{ a+b } ,\cfrac { bh }{ a+b } )\\ \therefore x=\cfrac { al }{ a+b } ,\quad y=\cfrac { bh }{ a+b } \\ AB=\sqrt { { l }^{ 2 }+{ h }^{ 2 } } =(a+b)\quad \quad \Rightarrow { l }^{ 2 }+{ h }^{ 2 }={ (a+b) }^{ 2 }\\ \therefore \cfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =\cfrac { { l }^{ 2 }+{ h }^{ 2 } }{ { (a+b) }^{ 2 } }$$$$\therefore \cfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$$   -required locus .Mathematics

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