MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
Menu
Hemanth Sai Grade: 11
        

a vertical line divides the triangle with vertices (0,0),(1,1),(9,1) in xy plane into 2 regions of equal area.the equation of line is x = ?

7 years ago

Answers : (1)

suryakanth AskiitiansExpert-IITB
105 Points
										

If x+ y+ z=0


Prove that:


(x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³ = 3(x²+xy+y²). (y²+yz+z²) (z²+zx+x²).


Answer:


Let a = (x²+xy+y²)


b= (y²+yz+z²)


c= (z²+zx+x²)


This equation reduces to proving that a3+b3+c3=3abc


This is possible if:



  • a+b+c = 0

  • or a=b=c


We realize by simple substitution(like taking x,y,z = (-1,0,1),(-2,0,2)) that a+b+c is not 0 all the times


Now considering a=b


If and only if


(x²+xy+y²) = (y²+yz+z²)


If and only if


x²+xy = yz+z²


i.e.,        x(x+y)=z(y+z)


Using the fact that x+y+z=0, we see this is nothing but,


                x(-z)=z(-x)


Hence a=b=c


=> a3+b3+c3=3abc


which is


(x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³ = 3(x²+xy+y²). (y²+yz+z²) (z²+zx+x²).


Hence proved.


Please feel free to post as many doubts on our discussion forum as you can. We are all IITians and here to help you in your IIT JEE preparation.



Win exciting gifts by answering the questions on Discussion Forum..



 



If x+ y+ z=0


Prove that:


(x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³ = 3(x²+xy+y²). (y²+yz+z²) (z²+zx+x²).


Answer:


Let a = (x²+xy+y²)


b= (y²+yz+z²)


c= (z²+zx+x²)


This equation reduces to proving that a3+b3+c3=3abc


This is possible if:


·         a+b+c = 0


·         or a=b=c


We realize by simple substitution that a+b+c is not 0 all the times


Now considering a=b


If and only if


(x²+xy+y²) = (y²+yz+z²)


If and only if


x²+xy = yz+z²


i.e.,        x(x+y)=z(y+z)


Using the fact that x+y+z=0, we see this is nothing but,


                x(-z)=z(-x)


Hence proved.


7 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies
  • Complete AIPMT/AIIMS Course and Test Series
  • OFFERED PRICE: Rs. 15,900
  • View Details
  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: Rs. 15,900
  • View Details

Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details