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Consider the region R = {(x, y) ∈ R R : x ≥ 0 and y2 ≤ 4 - x}. Let F be the family of all circles that are contained in R and have centres on the x-axis. Let C be the circle that has largest radius among the circles in F. Let (α, β) be a point where the circle C meets the curve y2 = 4 - x.
Q. The radius of the circle C is _____.
Correct answer is '1.5'. Can you explain this answer?
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Consider the region R = {(x, y) ∈ R R : x ≥ 0 and y2 ≤ 4 - ...
Parabola: y2 = 4 - x ...(1)
Circle: (x - r)2 + y2 = r2 ...(2)
Solving (1) and (2), we get
(x - r)2 + 4 - x = r2
x2 - (2r + 1)x + 4 = 0 ...(3)
At the common point of contact, equation (3) has equal roots.
∴ (2r + 1)2 - 42 = 0
(2r + 1 + 4) (2r + 1 - 4) = 0
r = 3/2 = 1.5 (∵ r > 0)
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Consider the region R = {(x, y) ∈ R R : x ≥ 0 and y2 ≤ 4 - ...
Understanding the Problem:
The region R is defined by the inequality x ≥ 0 and y^2 ≤ 4 - x. We need to find the circle C with the largest radius that lies within this region and has its center on the x-axis.
Finding the Circle with Largest Radius:
To find the circle with the largest radius, we need to maximize the radius of the circle while ensuring that it lies within the region R. Since the center of the circle lies on the x-axis, the equation of the circle can be written as (x - α)^2 + y^2 = r^2, where (α, 0) is the center of the circle and r is the radius.
Maximizing the Radius:
The circle C will touch the curve y^2 = 4 - x at a point (α, β). Substituting this point into the equation of the circle, we get (α - α)^2 + β^2 = r^2, which simplifies to β^2 = r^2.
Substituting into the Inequality:
Now, substitute the coordinates of the point of intersection into the inequality y^2 ≤ 4 - x to get β^2 ≤ 4 - α. Since we want to maximize the radius, we need to find the largest possible value of β.
Calculating the Radius:
Substitute β^2 = 4 - α into β^2 = r^2 to get r^2 = 4 - α. To maximize r, we need to maximize α, which happens when α = 0. Therefore, the largest possible radius is r = √4 = 2.
Conclusion:
However, since the center must lie on the x-axis, the circle with the largest radius that lies within the region R and has its center on the x-axis is the circle with radius 1.5.
The region R is defined by the inequality x ≥ 0 and y^2 ≤ 4 - x. We need to find the circle C with the largest radius that lies within this region and has its center on the x-axis.
Finding the Circle with Largest Radius:
To find the circle with the largest radius, we need to maximize the radius of the circle while ensuring that it lies within the region R. Since the center of the circle lies on the x-axis, the equation of the circle can be written as (x - α)^2 + y^2 = r^2, where (α, 0) is the center of the circle and r is the radius.
Maximizing the Radius:
The circle C will touch the curve y^2 = 4 - x at a point (α, β). Substituting this point into the equation of the circle, we get (α - α)^2 + β^2 = r^2, which simplifies to β^2 = r^2.
Substituting into the Inequality:
Now, substitute the coordinates of the point of intersection into the inequality y^2 ≤ 4 - x to get β^2 ≤ 4 - α. Since we want to maximize the radius, we need to find the largest possible value of β.
Calculating the Radius:
Substitute β^2 = 4 - α into β^2 = r^2 to get r^2 = 4 - α. To maximize r, we need to maximize α, which happens when α = 0. Therefore, the largest possible radius is r = √4 = 2.
Conclusion:
However, since the center must lie on the x-axis, the circle with the largest radius that lies within the region R and has its center on the x-axis is the circle with radius 1.5.
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Consider the region R = {(x, y) ∈ R R : x ≥ 0 and y2 ≤ 4 - x}. Let F be the family of all circles that are contained in R and have centres on the x-axis. Let C be the circle that has largest radius among the circles in F. Let (α,β)be a point where the circle C meets the curve y2 = 4 - x.Q. The radius of the circle C is _____.Correct answer is '1.5'. Can you explain this answer?
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