<p>If a circle passing through the point \((-1, 0)\) touches \(y\)-axis at (0, 2), then the length of the chord of the circle along the \(x\)-axis is</p>
Step-by-Step Solution
Key Concept: If a circle touches the y-axis at point (0, 2), its center must lie on the horizontal line y = 2, and the radius equals the x-coordinate of the center. Use the fact that the circle passes through (-1, 0) to find the center and radius.
<p><strong>Step 1:</strong> Since the circle touches the y-axis at (0, 2), the center must be at (h, 2) where h is the radius (distance from center to y-axis).</p><p><strong>Step 2:</strong> The circle passes through (-1, 0), so the distance from center (h, 2) to (-1, 0) equals the radius h.</p><p>$(h - (-1))^2 + (2 - 0)^2 = h^2$</p><p>$(h + 1)^2 + 4 = h^2$</p><p>$h^2 + 2h + 1 + 4 = h^2$</p><p>$2h + 5 = 0$</p><p>$h = -\frac{5}{2}$</p><p><strong>Step 3:</strong> Since h must be positive (center is to the right of y-axis for standard configuration), we take $h = \frac{5}{2}$, so center is at $(-\frac{5}{2}, 2)$ and radius = $\frac{5}{2}$.</p><p><strong>Step 4:</strong> For chord along x-axis (where y = 0), the distance from center $(-\frac{5}{2}, 2)$ to x-axis is 2.</p><p>Using chord formula: If d is perpendicular distance from center to chord, then chord length = $2\sqrt{r^2 - d^2}$</p><p>Chord length = $2\sqrt{(\frac{5}{2})^2 - 2^2} = 2\sqrt{\frac{25}{4} - 4} = 2\sqrt{\frac{9}{4}} = 2 \cdot \frac{3}{2} = 3$</p><p>∴ Answer: B (chord length = 3)</p>
Correct Answer: B