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Circle question

2021 · Shift 2 · Q26
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  5. /2021 · Shift 2 · Q26

Circle question

2021 · Shift 2 · Q26

JEE AdvancedMathematicsCircleNumerical+2 / −1
Consider the region R = {(x, y) ∈\in∈ R ×\times× R : x ≥\ge≥ 0 and y2 ≤\le≤ 4 −-− x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let (α\alphaα, β\betaβ) be a point where the circle C meets the curve y2 = 4 −-− x. The radius of the circle C is ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 1.50

Step-by-Step Solution

  1. Analyze the Region R The region R is defined by the inequalities x ≥ 0 and y2≤4−xy^2 ≤ 4 - xy2≤4−x.

    • x ≥ 0 restricts the region to the first and fourth quadrants (and the y-axis).
    • y2≤4−xy^2 ≤ 4 - xy2≤4−x can be rewritten as x≤4−y2x ≤ 4 - y^2x≤4−y2. This describes the region inside and on the parabola x=4−y2x = 4 - y^2x=4−y2. This parabola opens to the left, has its vertex at (4, 0), and intersects the y-axis at (0, 2) and (0, -2). So, the region R is bounded by the y-axis (x=0) on the left and the parabola x=4−y2x = 4 - y^2x=4−y2 on the right.

  2. Analyze the Family of Circles F The family F consists of all circles that are contained in R and have their centers on the x-axis. Let a circle C in this family have its center at (h, 0) and radius r. The equation of such a circle is (x−h)2+y2=r2(x - h)^2 + y^2 = r^2(x−h)2+y2=r2.

  3. Identify the Largest Circle C We are looking for the circle C with the largest possible radius. For a circle to have the maximum possible radius within a bounded region, it must be tangent to the boundaries of that region. In this case, the largest circle C must be tangent to both the y-axis (x = 0) and the parabola y2=4−xy^2 = 4 - xy2=4−x.

  4. Use the Tangency Conditions to Find the Radius

    • Tangency to the y-axis (x=0): The distance from the center of the circle (h, 0) to the line x = 0 is h. For the circle to be tangent to this line, this distance must be equal to the radius r. Since the circle is in the region x ≥ 0, its center (h, 0) must have h > 0. Therefore, h = r.

    • Substitute h = r into the circle's equation: The equation of the largest circle C becomes (x−r)2+y2=r2(x - r)^2 + y^2 = r^2(x−r)2+y2=r2.

    • Tangency to the parabola (y2=4−xy^2 = 4 - xy2=4−x): The circle must also be tangent to the parabola. To find the points of intersection between the circle and the parabola, we can solve their equations simultaneously. Substitute y2=4−xy^2 = 4 - xy2=4−x from the parabola's equation into the circle's equation: (x−r)2+(4−x)=r2(x - r)^2 + (4 - x) = r^2(x−r)2+(4−x)=r2 Expand and simplify this equation: x2−2rx+r2+4−x=r2x^2 - 2rx + r^2 + 4 - x = r^2x2−2rx+r2+4−x=r2 x2−(2r+1)x+4=0x^2 - (2r + 1)x + 4 = 0x2−(2r+1)x+4=0 This is a quadratic equation in x. For the circle and the parabola to be tangent, there must be exactly one real solution for x at the point of tangency. This means the discriminant (D) of the quadratic equation must be zero. D=b2−4ac=0D = b^2 - 4ac = 0D=b2−4ac=0 Here, a = 1, b = -(2r + 1), and c = 4. (−(2r+1))2−4(1)(4)=0(-(2r + 1))^2 - 4(1)(4) = 0(−(2r+1))2−4(1)(4)=0 (2r+1)2−16=0(2r + 1)^2 - 16 = 0(2r+1)2−16=0 (2r+1)2=16(2r + 1)^2 = 16(2r+1)2=16 Taking the square root of both sides: 2r+1=±42r + 1 = \pm 42r+1=±4

  5. Solve for the Radius r We have two possible cases:

    • Case 1: 2r + 1 = 4 2r = 3 r = 3/2 = 1.5

    • Case 2: 2r + 1 = -4 2r = -5 r = -5/2 = -2.5

    Since the radius r must be a positive value, we discard the second case. The only valid solution is r = 1.5.

  6. Conclusion The radius of the circle C, which has the largest radius among all circles in F, is 1.5. The information about the point (α, β) is extra detail not required to find the radius.

Final Answer: The radius of the circle C is 1.5.

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