Home
Class 12
MATHS
Prove that (x^2 + y^2)(z^2 + w^2) = X^2 ...

Prove that `(x^2 + y^2)(z^2 + w^2) = X^2 + Y^2 if (x + iy)(z + iw) = X + iY`

Text Solution

Verified by Experts

The correct Answer is:
ad
Now `(x + (1)/(x))^(20) = ""^(20)C_(0) x^(20) + ""^(20)C_(1)x^(18) + ""^(20)C_(2) x^(16) +""^(20)C_(3) x^(14) + …`
` + ""^(20)C_(9) x^(2) + ""^(20)C_(10) + ""^(20)C_(11) x^(-2) + …+ ""^(20)C_(20) x^(-20)`
` T_(r+1) = ""^(20)C_(r) . X^(20 - 2r) ` ...(i)
and `(x^(2) + 2 + (1)/(x^(2)))^(10) = =((1)/(x) + x)^(20) `
` = ""^(20)C_(0) x^(-20) + ""^(20)C_(1) x^(-18) + ""^(20)C_(2) x^(-16) `
` + ...+ ""^(20)C_(10) + ""^(20)C_(11)x^(2) + ""^(20)C_(12) x^(4)`
` + ...+ ""^(20)C_(20) x^(20)`
` therefore T_(6) = t_(5+1) = ""^(20)C_(5) x^(-10)` ...(ii)
According to the question , `""^(20)C_(r) = ""^(20)C_(5) `
` therefore r= 5 or 20 = r + 5 rArr Sr = 5 , 15 `
Promotional Banner

Topper's Solved these Questions

  • BIONOMIAL THEOREM

    ARIHANT MATHS|Exercise Exercise (Passage Based Questions)|21 Videos

  • BIONOMIAL THEOREM

    ARIHANT MATHS|Exercise Binomial Theorem Exerciese 4 : Single Integer Answer Type Questions|1 Videos

  • BIONOMIAL THEOREM

    ARIHANT MATHS|Exercise Exercise (Single Option Correct Type Questions)|30 Videos

  • AREA OF BOUNDED REGIONS

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|23 Videos

  • CIRCLE

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|16 Videos

Similar Questions

Explore conceptually related problems

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)| = (x-y)(y-z)(z-x)

Add x^(2) y ( y + z) + y ^(2) z ( y - z) and z ^(2) y ( y-z) - yz ( x ^(2) + y ^(2))

Prove that |(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y)^2(b-z)^2(c-x)^2(c-y)^2(c-z)^2|= |(1+a x)^2(1+b x)^2(1+c x)^2(1+a y)^2(1+b y)^2(1+c y)^2(1+a z)^2(1+b z)^2(1+c z)^2|=2(b-c)(c-c)(a-b)xx(y-z)(z-x)(x-y)dot

Prove that |[[x,y,z],[x^2,y^2,z^2],[x^3,y^3,z^3]]|= xyz (x-y)(y-z)(z-x)

If xy + yz + zx = 1 , prove that : x/(1-x^2)+y/(1-y^2)+z/(1-z^2)= (4xyz)/((1-x^2)(1-y^2)(1-z^2)) .

Add: 2x (z – x – y) and 2y (z – y – x)