An algebra problem by KSHITIZ CHAUHAN
let a , b , x , y be all real numbers such that a 2 + b 2 = 8 1 , x 2 + y 2 = 1 2 1 and a x + b y = 9 9 . then find the set of all possible values of a y − b x .
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
3 solutions
Wow, how did you know that identity? Any website or formulas page that has more algebraic formulas like this?
Log in to reply
Check this out: Brilliant : Advanced Factorization
KVPY 2011 question word to word!
This is cauchy Schwarz and we get a=9 and x=11 or b=9 and y=11
By the Cauchy-Schwarz Inequality,
a x + b y ≤ a 2 + b 2 ⋅ x 2 + y 2
Substituting the values given, we will see that the equation is hold.
The equation holds when
x a = y b
Cross-multiplying and subtracting a y from both sides,
a x − b y = 0
a^2+b^2=81 is an equation of circle with radius 9 units and centre at origin. Take a=9 Cos A , b=9 Sin A. Then the given straight line will be x Cos A + y Sin A=11 which is actually equation of the tangent to the bigger circle x^2 + y^2=121 at the point ( 11 Cos A, 11 Sin A). So x= 11 Cos A and y=11 Sin A. Thus ax-by will be equal to zero.
A geometric approach
Did the same way
Hi there, Sanchayapol Lewgasamsarn. I just want to tell you that I was able to verify your answer for your problem "Is It Even Possible?" Although I wasn't able to post my solution because I made too many mistakes. My solution used computer code. I don't want to reveal the secret here, but with the computer code I was able to find another way to do it that requires less than 3% of the work. It still remains to prove that the conjecture is true though without verifying it using the computer. Do you already have a good solution to this problem?
Log in to reply
Let me know if you already know what I'm talking about or if you would like me to elaborate through email communication. Thanks.
a^{2}+b^{2}=81 is an equation of circle with radius 9 units and centre at origin. Take a=9 Cos A , b=9 Sin A. Then the given straight line will be x Cos A + y Sin A=11 which is actually equation of the tangent to the bigger circle x^{2} + y^{2}=121 at the point ( 11 Cos A, 11 Sin A). So x= 11 Cos A and y=11 Sin A. Thus ax-by will be equal to zero
We know that ( a 2 + b 2 ) ( x 2 + y 2 ) = ( a x + b y ) 2 + ( a y − b x ) 2
Putting the given values, we see that, ( a y − b x ) 2 = 0 ⟹ a y − b x = 0