Question

The number of points \( (a, b) \), where \( a \) and \( b \) are positive integers, lying on the hyperbola \( x^2 - y^2 = 512 \) is:

• A. \( 3 \)
• B. \( 4 \)
• C. \( 5 \)
• D. \( 6 \)
• E. \( 7 \)

Hint:

For equations of the form \(x^2-y^2=N\), convert into factor pairs using \((x-y)(x+y)=N\). Then count valid factor pairs satisfying parity conditions.

Answer 1

The Correct Option is B

Concept: Using the identity \[ x^2-y^2=(x-y)(x+y) \] the equation becomes \[ (x-y)(x+y)=512 \] Let \[ x-y=u,\qquad x+y=v \] Then, \[ uv=512 \]

Step 1: Apply conditions for integer solutions.
Since \(x\) and \(y\) are positive integers, \[ x=\frac{u+v}{2}, \qquad y=\frac{v-u}{2} \] Therefore, \(u\) and \(v\) must both be even. Also, \[ v>u>0 \]

Step 2: Find factor pairs of \(512=2^9\).
Possible even factor pairs are: \[ (2,256),\ (4,128),\ (8,64),\ (16,32) \] These give: \[ (x,y)=(129,127) \] \[ (x,y)=(66,62) \] \[ (x,y)=(36,28) \] \[ (x,y)=(24,8) \] Thus, total positive integral solutions are: \[ 4 \] \[ \boxed{4} \]