Reverse fraction?

Algebra Level 3

If p a + q b + r c = 1 \dfrac{p}{a}+ \dfrac{q}{b}+\dfrac{r}{c}= 1 and a p + b q + c r = 0 \dfrac{a}{p}+\dfrac{b}{q}+\dfrac{c}{r}=0 then the value of p 2 a 2 + q 2 b 2 + r 2 c 2 = ? \dfrac{p^2}{a^2}+\dfrac{q^2}{b^2}+\dfrac{r^2}{c^2} =\space ?

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Aaryan Maheshwari by Aaryan Maheshwari , 16, India

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1 solution

X X
May 15, 2018

Let p a = x , q b = y , r c = z \frac pa=x,\frac qb=y,\frac rc=z ,we get x + y + z = 1 , 1 x + 1 y + 1 z = 0 , x y + y z + x z = 0 , x 2 + y 2 + z 2 = ( x + y + z ) 2 − 2 ( x y + y z + z x ) = 1 x+y+z=1,\frac1x+\frac1y+\frac1z=0,xy+yz+xz=0,x^2+y^2+z^2={(x+y+z)}^2-2(xy+yz+zx)=1

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