Question

If \( m \) is the slope and \( P(\beta, \beta) \) is the midpoint of a chord of contact of the circle \[ x^2 + y^2 = 125, \] then the number of values of \( \beta \) such that \( \beta \) and \( m \) are integers is: \

• A. \( 2 \)
• B. \( 4 \)
• C. \( 6 \)
• D. \( 8 \)
  \

Hint:

For a midpoint of a chord of contact, use the equation \( T = 0 \). The integer condition ensures that divisors of the circle's constant term are considered.

Answer 1

The Correct Option is C

Step 1: Equation of Chord of Contact
The equation of the given circle is: \[ x^2 + y^2 = 125. \] Using the midpoint formula for a chord of contact, the equation of the chord with midpoint \( (\beta, \beta) \) is: \[ T = 0, \quad \text{where } T = x\beta + y\beta - 125 = 0. \] \[ \beta x + \beta y = 125. \] Rewriting: \[ x + y = \frac{125}{\beta}. \] Step 2: Condition for Integer Values
Since \( \beta \) and \( m \) are integers, we require: \[ \frac{125}{\beta} \text{ to be an integer}. \] This means \( \beta \) must be a divisor of 125. Step 3: Finding Valid Values of \( \beta \)
The divisors of 125 are: \[ \pm 1, \pm 5, \pm 25, \pm 125. \] Since \( \beta \) is the midpoint coordinate of the chord, it must be an integer. The valid integer values of \( \beta \) are: \[ \pm 1, \pm 5, \pm 25. \] Thus, the number of possible values of \( \beta \) is: \[ 6. \] Step 4: Conclusion
Thus, the final answer is: \[ \boxed{6}. \] \bigskip