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Find the perimeter of the loop of the curve9aysquare =(x-2a) (x-5a)square
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Find the perimeter of the loop of the curve9aysquare =(x-2a) (x-5a)squ...
Perimeter of the Loop of a Parabola
The perimeter of the loop of a curve is the distance around the loop. In the case of a parabola, the loop refers to the portion of the curve that starts at the vertex and extends to the point where the curve intersects the x-axis again.
Equation of the Parabola
The given equation of the parabola is 9ay^2 = (x - 2a)(x - 5a)^2. This is a vertical parabola with its vertex at (3a, 0).
Finding the x-Intercepts
To find the points where the parabola intersects the x-axis, we set y = 0 in the equation and solve for x. We get:
0 = (x - 2a)(x - 5a)^2
x = 2a or x = 5a (double root)
Finding the Length of the Loop
The length of the loop is equal to the sum of the distances from the vertex to each x-intercept. Using the distance formula, we get:
d1 = sqrt[(2a - 3a)^2 + (0 - 0)^2] = a
d2 = 2sqrt[(5a - 3a)^2 + (0 - 0)^2] = 2a*sqrt(2)
Therefore, the perimeter of the loop is:
P = d1 + d2 + d1 = a + 2a*sqrt(2) + a = (2 + 2sqrt(2))a
Conclusion
The perimeter of the loop of the parabola 9ay^2 = (x - 2a)(x - 5a)^2 is (2 + 2sqrt(2))a.
The perimeter of the loop of a curve is the distance around the loop. In the case of a parabola, the loop refers to the portion of the curve that starts at the vertex and extends to the point where the curve intersects the x-axis again.
Equation of the Parabola
The given equation of the parabola is 9ay^2 = (x - 2a)(x - 5a)^2. This is a vertical parabola with its vertex at (3a, 0).
Finding the x-Intercepts
To find the points where the parabola intersects the x-axis, we set y = 0 in the equation and solve for x. We get:
0 = (x - 2a)(x - 5a)^2
x = 2a or x = 5a (double root)
Finding the Length of the Loop
The length of the loop is equal to the sum of the distances from the vertex to each x-intercept. Using the distance formula, we get:
d1 = sqrt[(2a - 3a)^2 + (0 - 0)^2] = a
d2 = 2sqrt[(5a - 3a)^2 + (0 - 0)^2] = 2a*sqrt(2)
Therefore, the perimeter of the loop is:
P = d1 + d2 + d1 = a + 2a*sqrt(2) + a = (2 + 2sqrt(2))a
Conclusion
The perimeter of the loop of the parabola 9ay^2 = (x - 2a)(x - 5a)^2 is (2 + 2sqrt(2))a.
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Find the perimeter of the loop of the curve9aysquare =(x-2a) (x-5a)square Related: Latus Rectum in Parabola,Examples (Part - 6) - Conic Sections, Mathematics, Class 11
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Find the perimeter of the loop of the curve9aysquare =(x-2a) (x-5a)square Related: Latus Rectum in Parabola,Examples (Part - 6) - Conic Sections, Mathematics, Class 11 for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Find the perimeter of the loop of the curve9aysquare =(x-2a) (x-5a)square Related: Latus Rectum in Parabola,Examples (Part - 6) - Conic Sections, Mathematics, Class 11 covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the perimeter of the loop of the curve9aysquare =(x-2a) (x-5a)square Related: Latus Rectum in Parabola,Examples (Part - 6) - Conic Sections, Mathematics, Class 11.
Find the perimeter of the loop of the curve9aysquare =(x-2a) (x-5a)square Related: Latus Rectum in Parabola,Examples (Part - 6) - Conic Sections, Mathematics, Class 11 for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Find the perimeter of the loop of the curve9aysquare =(x-2a) (x-5a)square Related: Latus Rectum in Parabola,Examples (Part - 6) - Conic Sections, Mathematics, Class 11 covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the perimeter of the loop of the curve9aysquare =(x-2a) (x-5a)square Related: Latus Rectum in Parabola,Examples (Part - 6) - Conic Sections, Mathematics, Class 11.
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