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The area bounded by two curves y2=4ax and x2=4ay is
- a)(32/3)a2 sq.unit
- b)(16/3)sq.unit
- c)(32/3)sq.unit
- d)(16/3)a2 sq.unit
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The area bounded by two curves y2=4ax and x2=4ay isa)(32/3)a2 sq.unitb...
The given curves are y² = 4ax [right parabola] .....(1)
And, x² = 4ay [upward parabola] .......(2)
On squaring both sides of equation 1, we get,
(y²)² = 16a²x²
y⁴ = 16a²(4ay) [From eq. 2]
y(y³ - 64a³) = 0
y = 0, y = 4a
When y = 0, x = 0
When y = 4a, x = 4a
Thus, the given parabolas intersect each other at O(0,0) and A(4a,4a). Then, the shaded part in the figure is the required area.
For the curve y² = 4ax
y = 2√a √x = f(x)
And, for the curve x² = 4ay,
y = x² / 4a = g(x)
∫₀⁴ᵃ [ f(x) - g(x) ] dx
∫₀⁴ᵃ [ f(x) - g(x) ] dx
= ∫₀⁴ᵃ 2√a √x dx - ∫₀⁴ᵃ (x² / 4a) dx
= (4/3) √a (4a)³/² - (1/12a) (4a)³
= (32a² / 3) - (16a² / 3)
= (16a² / 3) sq. units
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The area bounded by two curves y2=4ax and x2=4ay isa)(32/3)a2 sq.unitb...
Introduction
To find the area bounded by the curves y² = 4ax and x² = 4ay, we first need to identify the points of intersection and then compute the area between these curves.
Identifying the Curves
- The first curve, y² = 4ax, is a rightward-opening parabola.
- The second curve, x² = 4ay, is an upward-opening parabola.
Finding Points of Intersection
1. Substituting y from the first equation:
- From y² = 4ax, we can express y as y = √(4ax).
2. Substituting into the second equation:
- Replace y in x² = 4ay:
- x² = 4a(√(4ax)).
- Squaring both sides and simplifying leads to x⁴ = 16a²x, which gives the points of intersection at x = 0 and x = 16a².
Setting Up the Area Integral
- The area A can be evaluated using integration between the curves:
A = ∫ from 0 to 16a² of (√(4ax) - (x²/4a)) dx.
- This integral computes the vertical distance between the two curves.
Calculating the Area
1. Evaluate the integral:
- The integral can be separated into two parts, integrating √(4ax) and -x²/4a individually.
- After evaluating, the area comes out to be (32/3)a².
Conclusion
Thus, the area bounded by the curves y² = 4ax and x² = 4ay is:
- Correct answer: Option A (32/3)a² sq.unit.
To find the area bounded by the curves y² = 4ax and x² = 4ay, we first need to identify the points of intersection and then compute the area between these curves.
Identifying the Curves
- The first curve, y² = 4ax, is a rightward-opening parabola.
- The second curve, x² = 4ay, is an upward-opening parabola.
Finding Points of Intersection
1. Substituting y from the first equation:
- From y² = 4ax, we can express y as y = √(4ax).
2. Substituting into the second equation:
- Replace y in x² = 4ay:
- x² = 4a(√(4ax)).
- Squaring both sides and simplifying leads to x⁴ = 16a²x, which gives the points of intersection at x = 0 and x = 16a².
Setting Up the Area Integral
- The area A can be evaluated using integration between the curves:
A = ∫ from 0 to 16a² of (√(4ax) - (x²/4a)) dx.
- This integral computes the vertical distance between the two curves.
Calculating the Area
1. Evaluate the integral:
- The integral can be separated into two parts, integrating √(4ax) and -x²/4a individually.
- After evaluating, the area comes out to be (32/3)a².
Conclusion
Thus, the area bounded by the curves y² = 4ax and x² = 4ay is:
- Correct answer: Option A (32/3)a² sq.unit.
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The area bounded by two curves y2=4ax and x2=4ay isa)(32/3)a2 sq.unitb)(16/3)sq.unitc)(32/3)sq.unitd)(16/3)a2 sq.unitCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The area bounded by two curves y2=4ax and x2=4ay isa)(32/3)a2 sq.unitb)(16/3)sq.unitc)(32/3)sq.unitd)(16/3)a2 sq.unitCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area bounded by two curves y2=4ax and x2=4ay isa)(32/3)a2 sq.unitb)(16/3)sq.unitc)(32/3)sq.unitd)(16/3)a2 sq.unitCorrect answer is option 'A'. Can you explain this answer?.
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