If a - b - c - p - q - r are three complex numbers such that p - a - q - b - r - c -1- i and a - p - b - q - c - r -0 then value of p 2- a 2- q 2- b 2- r 2- c 2 isA- 2 iB- -2 iC- 0D- -1

P/a+q/b+r/c=1+i, ⇒(p/a+q/b+r/c)^2=(1+i)^2=2i ⇒ p^2/a^2+q^2/b^2+r^2/c^2+2(qr/bc+rp/ca+pq/ab)=2i ⇒ ⇒ p^2/a^2+q^2/b^2+r^2/c^2+2abc/pqr(a/p+b/q+c/r)=2i ⇒ p^ ...

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Byju's Answer

Standard XII

Mathematics

Integration by Parts

If a , b , c ...

Question

If $a, b, c, p, q, r$ are three complex numbers such that $\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1 + i$ and $\frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0$ then value of $\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}}$ is

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Solution

The correct option is C 2i
$\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1 + i, \Rightarrow {(\frac{p}{a} + \frac{q}{b} + \frac{r}{c})}^{2} = (1 + i)^{2} = 2 i$

$\Rightarrow \frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}} + 2 (\frac{q r}{b c} + \frac{r p}{c a} + \frac{p q}{a b}) = 2 i \Rightarrow$

$\Rightarrow \frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}} + \frac{2 a b c}{p q r} (\frac{a}{p} + \frac{b}{q} + \frac{c}{r}) = 2 i$

$\Rightarrow \frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}} = 2 i$

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Similar questions

Q. If

a, b, c, p, q, r

are three non-zero complex number such that

\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1 + i

and

\frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0

, then value of

\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}}

Q. If

a, b, c, p, q, r

are thre non-zero complex numbers such that

\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1 + i

and

\frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0,

then value of

\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}}

Q. If

\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1

and

\frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0

then value of

\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}}

is.

Q. If

α, β, γ

and

a, b, c

are the complex numbers such that

\frac{α}{a} + \frac{β}{b} + \frac{γ}{c} = 1 + i

and

\frac{a}{α} + \frac{b}{β} + \frac{c}{γ} = 0

such that

\frac{α^{2}}{a^{2}} + \frac{β^{2}}{b^{2}} + \frac{γ^{2}}{c^{2}} = p + i q

where

p, q \in R

then the value of

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Q. If

p = - 2, q = 1

and

r = 3

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If a,b,c,p,q,r are three complex numbers such that pa+qb+rc=1+i and ap+bq+cr=0 then value of p2a2+q2b2+r2c2 is

The correct option is C 2ipa+qb+rc=1+i, ⇒(pa+qb+rc)2=(1+i)2=2i ⇒ p2a2+q2b2+r2c2+2(qrbc+rpca+pqab)=2i ⇒ ⇒ p2a2+q2b2+r2c2+2abcpqr(ap+bq+cr)=2i ⇒ p2a2+q2b2+r2c2=2i

If $a, b, c, p, q, r$ are three complex numbers such that $\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1 + i$ and $\frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0$ then value of $\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}}$ is