Through the point $$P($$α,β$$)$$, where αβ$$0$$ the straight line $$xa+yb=1$$ is drawn so as to form with coordinateaxes a triangle of area S. If αβ$$0$$, then the least value of S is

Question

Through the point $$P($$α,β$$)$$, where αβ>$$0$$ the straight line $$xa+yb=1$$ is drawn so as to form with coordinateaxes a triangle of area S. If αβ>$$0$$, then the least value of S is

Infinity Learn · Accepted Answer

The equation of the given line $$isxa+yb=1$$                                         (1)This$$ line cuts $$x-axis$$ and $$y-axis$$ at $$A(a$$, $$0) and $$B(0$$, $$b)$$ respectively.Since area of ∆OAB $$=$$ S (Given)∴$$12ab=S$$  or $$ab=2S$$ ∵ab > $$0$$      (2)Since$$ the line $$(1) passes through the point $$P($$α,β$$)$$∴α$$a+$$β$$b=1$$  or  α$$a+$$αβ$$2S=1$$     [Using (2)]or  α$$2$$β$$-2aS+2$$α$$S=0Since$$ a is real, ∴$$4S2-8$$αβS≥$$0or$$  $$4S2$$≥$$8$$αβS  or  S≥$$2$$αβ    ∵$$S=12ab$$>$$0$$  as  ab > $$0Hence$$ the least value of $$S=2$$αβ

Through the point $P (α, β)$ , where $α β > 0$ the straight line $\frac{x}{a} + \frac{y}{b} = 1$ is drawn so as to form with coordinate
axes a triangle of area S. If $α β > 0$ , then the least value of S is

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Through the point P(α,β), where αβ>0 the straight line xa+yb=1 is drawn so as to form with coordinateaxes a triangle of area S. If αβ>0, then the least value of S is

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Ready to Test Your Skills?

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Through the point $P (α, β)$ , where $α β > 0$ the straight line $\frac{x}{a} + \frac{y}{b} = 1$ is drawn so as to form with coordinate
axes a triangle of area S. If $α β > 0$ , then the least value of S is