Question

If a point \(P(\alpha, \beta)\) on the line \(y = 1\) is such that the two distinct chords drawn on \(x^2 + y^2 - \alpha x - y = 0\) from \(P\) are bisected by the x-axis, then

• A. \(\alpha^2 \lt 8\)
• B. \(\alpha = 2\sqrt{2}\)
• C. \(\alpha^2 \gt 8\)
• D. \(\alpha = -2\sqrt{2}\)

Hint:

For a chord bisected by a line, use the midpoint formula and intersection with circle to set up condition on parameters.

Answer 1

The Correct Option is C

Step 1: Equation of chord with midpoint on x-axis.
If midpoint of chord is \((h,0)\), then chord equation using midpoint formula: \(T = S_1\) gives \[ xx_1 + yy_1 - \frac{\alpha}{2}(x + x_1) - \frac{1}{2}(y + y_1) = 0 \]

Step 2: Condition for P on line y=1.
Point \(P(\alpha,1)\) lies on such chords. Solve quadratic in x for intersection with circle.

Step 3: Chord bisected by x-axis.
If chord is bisected by x-axis, product of roots positive. Condition reduces to \[ \alpha^2 - 8 \gt 0 \]

Step 4: Solve inequality.
\[ \alpha^2 \gt 8 \]

Step 5: Final conclusion.
Hence, \[ \boxed{\alpha^2 \gt 8} \]