Question

Let \( C \) be the circle of minimum area touching the parabola \( y = 6 - x^2 \) and the lines \( y = \sqrt{3} |x| \). Then, which one of the following points lies on the circle \( C \)?

• A. \( (2, 4) \)
• B. \( (1, 2) \)
• C. \( (2, 2) \)
• D. \( (1, 1) \)

Answer 1

The Correct Option is A

The equation of the circle is:

\[ x^2 + (y - (6 - r))^2 = r^2, \]

where the center is \((0, 6 - r)\) and radius is \(r\).

Step 1: Condition for tangency with \(y = \sqrt{3}|x|\): The perpendicular distance from the center \((0, 6 - r)\) to the line \(y = \sqrt{3}|x|\) must equal the radius \(r\).
For the line \(y = \sqrt{3}x\), the distance is: \[ \frac{|0 - (6 - r)|}{\sqrt{1^2 + (\sqrt{3})^2}} = r. \] Simplify: \[ \frac{|6 - r|}{2} = r. \]

Step 2: Solve for \(r\):
Case 1: \(6 - r = 2r \implies 6 = 3r \implies r = 2.\)
Case 2: \(6 - r = -2r \implies 6 = -r \implies r = -6\) (not valid as \(r > 0\)).
Hence, \(r = 2\).

Step 3: Equation of the circle: Substituting \(r = 2\), the center becomes \((0, 6 - 2) = (0, 4)\).
The equation of the circle is: \[ x^2 + (y - 4)^2 = 4. \]

Step 4: Check which point lies on the circle: Substituting \((2, 4)\) into the equation:
\[ 2^2 + (4 - 4)^2 = 4 \implies 4 + 0 = 4. \] Thus, \((2, 4)\) lies on the circle.

Answer 2

To solve the problem of identifying the point lying on the smallest circle that touches the parabola \( y = 6 - x^2 \) and the lines \( y = \sqrt{3} |x| \), we need to analyze the geometric positioning and the constraint on the circle.

1. Understand the geometric constraints:
   • The parabola \( y = 6 - x^2 \) is open downward with vertex at \( (0, 6) \).
   • The lines \( y = \sqrt{3} |x| \) form a V-shape with vertex at the origin and slopes of \(\sqrt{3}\) and \(-\sqrt{3}\).
2. Visualize the circle constraint:
   • The circle must be tangent to the parabola and the two lines. This implies that it should be contained in the region bounded by these curves.
   • A circle touching all these curves at minimum size will likely touch the vertex or near-vertex regions of these curves.
3. Determine the point the circle passes through:
   • Based on geometric symmetry and considering candidate points, evaluate which point lies on or very close to one of the parabola's or lines' tangents near their vertices.
   • \((2, 4)\) is a plausible candidate because the parabola at \( x = 2 \) gives us \( y = 6 - (2)^2 = 2 \). Thus, the point \((2, 4)\) potentially allows tangency or close proximity to both the parabola and the lines.

After analyzing the options, it turns out that the point \( (2, 4) \) satisfies the tangency condition for the circle touching both described curves in the context of enclosing it within the arcs of the parabola and the lines.

Hence, the correct answer is \( (2, 4) \).