JEE Exam > JEE Questions > Consider the region R = {(x, y) ∈ R R : ...
Start Learning for Free
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?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
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)
Free Test
FREE
| Start Free Test |
Community Answer
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.
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Top Courses for JEEView all
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?
Question Description
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? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about 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? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about 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? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
Solutions for 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? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of 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? defined & explained in the simplest way possible. Besides giving the explanation of
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?, a detailed solution for 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? has been provided alongside types of 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? theory, EduRev gives you an
ample number of questions to practice 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? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup