<p>A circle \(C\) passes through the points of intersection of the parabola \(y + 1 = (x - 4)^2\) and the x-axis. The length of tangent from origin to \(C\) is:</p>
Step-by-Step Solution
Key Concept: Find the intersection points of the parabola with the x-axis, then determine the circle passing through these points. Use the tangent length formula from an external point to a circle.
<p><strong>Step 1: Find intersection points of parabola with x-axis</strong></p><p>The parabola is $y + 1 = (x-4)^2$. Setting $y = 0$: $1 = (x-4)^2$</p><p>$(x-4)^2 = 1 \Rightarrow x - 4 = \pm 1$</p><p>$x = 5$ or $x = 3$</p><p>Intersection points: $A(3, 0)$ and $B(5, 0)$</p><p><strong>Step 2: Set up the circle equation</strong></p><p>Since the circle passes through $(3, 0)$ and $(5, 0)$ on the x-axis, the chord $AB$ is horizontal. The center must lie on the perpendicular bisector of $AB$, which is the vertical line $x = 4$.</p><p>Let the center be $C(4, k)$ for some $k$ to be determined.</p><p><strong>Step 3: Use the condition that the circle passes through A and B</strong></p><p>Radius $r = \sqrt{(4-3)^2 + (k-0)^2} = \sqrt{1 + k^2}$</p><p>We can verify: distance from $C(4,k)$ to $B(5,0)$ is also $\sqrt{1 + k^2}$ ✓</p><p><strong>Step 4: Apply the tangent length formula</strong></p><p>For a tangent from origin $O(0,0)$ to circle with center $C(4, k)$ and radius $r$:</p><p>Tangent length $= \sqrt{d^2 - r^2}$, where $d = |OC|$</p><p>$d = \sqrt{4^2 + k^2} = \sqrt{16 + k^2}$</p><p>Tangent length $= \sqrt{(16 + k^2) - (1 + k^2)} = \sqrt{15}$</p><p><strong>Note:</strong> The value of $k$ cancels out, showing this is independent of where exactly the center lies on $x = 4$.</p><p><strong>∴ Answer: d</strong></p>
Correct Answer: d