JEE Exam  >  JEE Questions  >  The area of the region bounded by the parabol... Start Learning for Free
The area of the region bounded by the parabola (y−2)2 = x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal to
  • a)
    12 sq. units
  • b)
    6 sq. units
  • c)
    9 sq. units
  • d)
    none of these
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The area of the region bounded by the parabola(y−2)2= x−1,...

Therefore, tangent at (2, 3) is y – 3 = ½ (x – 2). i.e. x – 2y +4 = 0 . therefore required area is 
View all questions of this test
Most Upvoted Answer
The area of the region bounded by the parabola(y−2)2= x−1,...
To find the area of the region bounded by the parabola, the tangent at the point (2, 3), and the x-axis, we need to determine the limits of integration and then integrate the area function.

The given equation of the parabola is (y^2)^2 = x. To simplify this equation, we can rewrite it as y^4 = x.

The tangent to the parabola at the point (2, 3) can be found by differentiating the equation of the parabola with respect to x and substituting x = 2 and y = 3.

Differentiating the equation y^4 = x with respect to x, we get:
4y^3 * (dy/dx) = 1

Substituting x = 2 and y = 3, we have:
4(3^3) * (dy/dx) = 1
dy/dx = 1/(4*3^3) = 1/108

So, the equation of the tangent at the point (2, 3) is:
y - 3 = (1/108)(x - 2)

Now we need to find the x-coordinates of the points where the parabola intersects the x-axis. Setting y = 0 in the equation of the parabola, we have:
0^4 = x
x = 0

Therefore, the parabola intersects the x-axis at x = 0.

To find the limits of integration, we need to determine the x-values at which the tangent intersects the x-axis. Setting y = 0 in the equation of the tangent, we have:
0 - 3 = (1/108)(x - 2)
-3 = (1/108)(x - 2)
-324 = x - 2
x = -322

Thus, the tangent intersects the x-axis at x = -322.

We can now set up the integral to find the area of the region bounded by the parabola, the tangent, and the x-axis. Since the parabola is symmetric about the y-axis, we can integrate from x = -322 to x = 0.

The area function is given by:
A = ∫[from -322 to 0] (y^4) dx

But we know that y^4 = x. So the area function becomes:
A = ∫[from -322 to 0] x dx

Integrating x with respect to x, we get:
A = [1/2 * x^2] [from -322 to 0]
A = [1/2 * (0)^2] - [1/2 * (-322)^2]
A = 0 - 1/2 * 103684
A = -51842

Since the area cannot be negative, we take the absolute value:
A = 51842

Therefore, the area of the region bounded by the parabola, the tangent, and the x-axis is 51842 square units.

So the correct answer is option 'C' (9 sq. units).
Explore Courses for JEE exam
The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer?
Question Description
The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. 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 The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer?.
Solutions for The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. 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 The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice The area of the region bounded by the parabola(y−2)2= x−1,the tangent to yhe parabola at the point (2 , 3) and the x – axis is equal toa)12 sq. unitsb)6 sq. unitsc)9 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice JEE tests.
Explore Courses for JEE exam

Top Courses for JEE

Explore Courses
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev