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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 value of α is _____.
Correct answer is '2'. Can you explain this answer?
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Consider the region R = {(x, y) ∈ R R : x ≥ 0 and y2 ≤ 4 - ...
However, I can provide a description of the region R and some properties.
The region R can be described as the set of all points (x, y) such that x is greater than or equal to 0 and less than or equal to pi/2, and y is greater than or equal to 0 and less than or equal to sin(x).
Geometrically, this region is a portion of the curve y = sin(x) that lies above the x-axis and is bounded by the x-axis and the vertical line x = pi/2.
Some properties of this region include:
- It is a bounded region, meaning it has finite size and is contained within a finite area of the xy-plane.
- It is a connected region, meaning any two points within the region can be connected by a continuous path that lies entirely within the region.
- It has a finite area, which can be found by integrating the function y = sin(x) over the interval [0, pi/2]. The area is equal to 1.
The region R can be described as the set of all points (x, y) such that x is greater than or equal to 0 and less than or equal to pi/2, and y is greater than or equal to 0 and less than or equal to sin(x).
Geometrically, this region is a portion of the curve y = sin(x) that lies above the x-axis and is bounded by the x-axis and the vertical line x = pi/2.
Some properties of this region include:
- It is a bounded region, meaning it has finite size and is contained within a finite area of the xy-plane.
- It is a connected region, meaning any two points within the region can be connected by a continuous path that lies entirely within the region.
- It has a finite area, which can be found by integrating the function y = sin(x) over the interval [0, pi/2]. The area is equal to 1.
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Consider the region R = {(x, y) ∈ R R : x ≥ 0 and y2 ≤ 4 - ...
Let the equation of the circle be (x - r)2 + y2 = r2.
Equation of the parabola: y2 = 4 - x
Solving them,
(x - r)2 + 4 - x = r2
⇒ x2 - x(2r + 1) + 4 = 0
Since the circle and the parabola meet tangentially;
(2r + 1)2 - 16 = 0
⇒ 2r + 1 = 4
⇒ r = 3/2 and x2 - 4x + 4 = 0
(x - 2)2 = 0
⇒ α = 2, r = 3/2 = 1.5
Equation of the parabola: y2 = 4 - x
Solving them,
(x - r)2 + 4 - x = r2
⇒ x2 - x(2r + 1) + 4 = 0
Since the circle and the parabola meet tangentially;
(2r + 1)2 - 16 = 0
⇒ 2r + 1 = 4
⇒ r = 3/2 and x2 - 4x + 4 = 0
(x - 2)2 = 0
⇒ α = 2, r = 3/2 = 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 value of α is _____.Correct answer is '2'. Can you explain this answer?
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