JEE Exam > JEE Questions > The subnormal to the curve xy = c2 at any poi...
Start Learning for Free
The subnormal to the curve xy = c2 at any point varies directly as
- a)Cube of the ordinate(y3)
- b)Square of ordinate (y2)
- c)Ordinate (y)
- d)None
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The subnormal to the curve xy = c2 at any point varies directly asa) ...
Statement: The subnormal to the curve xy = c^2 at any point varies directly as the cube of the ordinate(y^3).
Explanation:
To understand the solution, we need to know what subnormal is, and how it is related to the given curve.
Subnormal: The line segment drawn from the point of contact on the curve to the x-axis, perpendicular to the tangent at that point is called the subnormal.
In the given curve, let P(x, y) be any point on the curve. Then, the equation of the curve can be written as y = c^2/x.
Now, let us find the slope of the tangent to the curve at point P.
Differentiating y = c^2/x with respect to x, we get
(dy/dx) * x = -c^2/x^2
=> dy/dx = -c^2/x^3
Therefore, the slope of the tangent at point P is -c^2/x^3.
Now, let us find the equation of the tangent at point P.
Equation of the tangent at point P can be written as
y - y1 = m(x - x1)
=> y - c^2/x = -c^2/x^3(x - x)
=> y = -c^2/x^2 + 2c^2/x
Now, let us find the subnormal at point P.
The subnormal is perpendicular to the tangent and passes through the point P.
Let Q be the point of contact of the subnormal with the curve. Then, the equation of the subnormal can be written as
(y - y1) = -1/m(x - x1)
=> y - c^2/x = x^3/c^2
=> y = x^3/c^2 + c^2/x
Now, we need to show that the subnormal varies directly as the cube of the ordinate.
Let y = kx^3/c^2 + c^2/x, where k is a constant.
Differentiating y with respect to x, we get
(dy/dx) = 3kx^2/c^2 - c^2/x^2
Therefore, the slope of the subnormal is (dy/dx) = 3kx^2/c^2 - c^2/x^2.
Now, we know that the subnormal is perpendicular to the tangent at point P. Therefore, the product of their slopes is -1.
=> (-c^2/x^3) * (3kx^2/c^2 - c^2/x^2) = -1
=> 3kx^4 - c^4 = -x^5
=> k = -1/3c^4
Therefore, y = (-1/3c^4) * x^3/c^2 + c^2/x
=> y = -x^3/(3c^6) + c^2/x
Hence, we can see that the subnormal varies directly as the cube of the ordinate.
Therefore, the correct answer is option 'A'.
Explanation:
To understand the solution, we need to know what subnormal is, and how it is related to the given curve.
Subnormal: The line segment drawn from the point of contact on the curve to the x-axis, perpendicular to the tangent at that point is called the subnormal.
In the given curve, let P(x, y) be any point on the curve. Then, the equation of the curve can be written as y = c^2/x.
Now, let us find the slope of the tangent to the curve at point P.
Differentiating y = c^2/x with respect to x, we get
(dy/dx) * x = -c^2/x^2
=> dy/dx = -c^2/x^3
Therefore, the slope of the tangent at point P is -c^2/x^3.
Now, let us find the equation of the tangent at point P.
Equation of the tangent at point P can be written as
y - y1 = m(x - x1)
=> y - c^2/x = -c^2/x^3(x - x)
=> y = -c^2/x^2 + 2c^2/x
Now, let us find the subnormal at point P.
The subnormal is perpendicular to the tangent and passes through the point P.
Let Q be the point of contact of the subnormal with the curve. Then, the equation of the subnormal can be written as
(y - y1) = -1/m(x - x1)
=> y - c^2/x = x^3/c^2
=> y = x^3/c^2 + c^2/x
Now, we need to show that the subnormal varies directly as the cube of the ordinate.
Let y = kx^3/c^2 + c^2/x, where k is a constant.
Differentiating y with respect to x, we get
(dy/dx) = 3kx^2/c^2 - c^2/x^2
Therefore, the slope of the subnormal is (dy/dx) = 3kx^2/c^2 - c^2/x^2.
Now, we know that the subnormal is perpendicular to the tangent at point P. Therefore, the product of their slopes is -1.
=> (-c^2/x^3) * (3kx^2/c^2 - c^2/x^2) = -1
=> 3kx^4 - c^4 = -x^5
=> k = -1/3c^4
Therefore, y = (-1/3c^4) * x^3/c^2 + c^2/x
=> y = -x^3/(3c^6) + c^2/x
Hence, we can see that the subnormal varies directly as the cube of the ordinate.
Therefore, the correct answer is option 'A'.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
The subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct answer is option 'A'. Can you explain this answer?
Question Description
The subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct 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 subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct 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 subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct answer is option 'A'. Can you explain this answer?.
The subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct 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 subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct 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 subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct answer is option 'A'. Can you explain this answer?.
Solutions for The subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct answer is option 'A'. 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 subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct answer is option 'A'. Can you explain this answer?, a detailed solution for The subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of The subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The subnormal to the curve xy = c2 at any point varies directly asa) Cube of the ordinate(y3) b) Square of ordinate (y2) c) Ordinate (y) d) None Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup