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The no. of ordered pairs of integers (x,y) for which 9x 2 -6x + y 2 = 24 is= ?

The no. of ordered pairs of integers (x,y) for which 9x2-6x + y2 = 24 is= ?

Grade:12

3 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Basit,

 

The given eqn can be written as (3x-1)2+y2=52,

32+42=52 is the only way 25 can be written as sum of sq of two integers.

 

So either 3x-1 = +/- 3or4

and y = +/- 3or4

 

Solve the equations, and you can see (-1,3) (-1,-3) are the only pairs of integers satisfying the equation.

So there are two ordered pairs.

 

Hope that helps.

 

Best Regards,

Ashwin (IIT Madras).

basit ali
41 Points
9 years ago

I liked that but the ans is 3 ordered pairs.

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Basit,

 

Absolutely.... yeah it should be three ordered pairs.

I'd taken only the Phythogorain Triplets....

The most Trivial one, I'd ignored.

 

If M2+N2 = L2, then M=0,N=L is also a set....

 

So 3x-1 = 5 and y=0 is another trivial possibility.

x=2, y=0.... which is the other ordered pair (2,0).

 

So total of 3 ordered pairs.

 

Regards,

Ashwin (IIT Madras).

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