Question

ICSE Mathematics for Class 9
(b) If the sum of the squares of three numbers is equal to the square of their sum, prove that the
sum of the products of the three numbers taking two at a time is equal to zero.​

Answer 1

Let the numbers be a,b and c

Given,

$a^2+b^2+c^2=(a+b+c)^2$

We know that,

$(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx$

Thus,

$a^2+b^2+c^2=a^2+b^2+c^2+2(ab+bc+ca)\\2(ab+bc+ca)=0$

As

• 2 can never be zero.
• the product of two numbers be zero only if one of them is zero.

$ab+bc+ca=0$

Hence proved.

Note:-

"Sum of the products of the three numbers taking two at a time is equal to zero" means first take any two numbers multiply them and repeat this process until no more possible pairs are left and finally add them.

For Example,

"Sum of the products of the Four numbers taking two at a time is equal to zero"

For a,b,c,d

ab+ac+ad+bc+bd+cd

Answer 2

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Question :-

If the sum of the squares of three numbers is equal to the square of their sum, prove that the

sum of the products of the three numbers taking two at a time is equal to zero.

let 3 no.s be a , b , c

Given :-

$(a + b + c )^2 = a ^2 + b ^2 + c^2$----------(i)

To prove :-

If :-

$(a + b + c )^2 = a ^2 + b ^2 + c^2$

then ;

ab + bc + ca = 0

formulae used :-

$(a + b + c )^2 = a ^2 + b ^2 + c^2+2(ab + bc + ca)$

step by step explanation :-

let

$(a + b + c )^2 = a ^2 + b ^2 + c^2 = x$

x = x + 2(ab+bc+ca)

2(ab + bc + ca )= x - x

ab + bc + ca = 0/2

ab + bc + ca = 0

.............Hence proved

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