Question
If the relation between subnormal 'SN' and subtangent 'ST' at any point on the curve by is
then
Hint:
We are given a curve. We are given the relation between the subtangent and the subnormal. We have to find the ratio of the variables p and q. We will first find the values of subtangent and subnormal using the formulae. Then we will find the ratio.
The correct answer is: ![fraction numerator 8 b over denominator 27 end fraction](data:image/png;base64,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)
The given curve is
.
The relation between subtangent "ST" and subnormal is "SN" is p(SN) = q(ST)2.
We have to find the ratio of
.
Subtangent: It is the projection of tangent on x-axis from the x intercept to point of tangency.
Subnormal: It is the projection of normal on x-axis from the x intercept to point of tangency.
The formula of subtangent is ST= ![fraction numerator y over denominator fraction numerator d y over denominator d x end fraction end fraction](data:image/png;base64,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)
The formula of subnormal is SN = ![y fraction numerator d y over denominator d x end fraction](data:image/png;base64,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)
we will differentiate the curve w.r.t x
![y squared space equals left parenthesis x space plus space a right parenthesis cubed space
2 y fraction numerator d y over denominator d x end fraction equals space 3 left parenthesis x space plus space a right parenthesis squared left parenthesis 1 right parenthesis space space space space space space space
space space space fraction numerator d y over denominator d x end fraction space equals fraction numerator 3 left parenthesis x space plus space a right parenthesis squared over denominator 2 y end fraction
N o w comma space S T space equals space fraction numerator y over denominator begin display style fraction numerator 3 left parenthesis x space plus space a right parenthesis squared over denominator 2 y end fraction end style end fraction
space space space space space space space space space space space space space space space equals fraction numerator 2 y squared over denominator 3 left parenthesis x space plus space a right parenthesis squared end fraction
space space space space space space space space space space space space space space equals fraction numerator 2 left parenthesis x space plus space a right parenthesis cubed over denominator 3 left parenthesis x space plus space a right parenthesis squared end fraction
space space space space space space space space space space space space space space space equals space 2 over 3 left parenthesis x space plus space a right parenthesis
left parenthesis S T right parenthesis to the power of 2 space end exponent equals 4 over 9 left parenthesis x space plus space a right parenthesis squared](data:image/png;base64,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)
![S N space equals space y space cross times fraction numerator 3 left parenthesis x space plus space a right parenthesis squared over denominator 2 y end fraction
space space space space space space equals 3 over 2 left parenthesis x space plus space a right parenthesis squared](data:image/png;base64,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)
Now, p(SN) = q(ST)2
![p space cross times space 3 over 2 left parenthesis x space plus space a right parenthesis squared equals q cross times fraction numerator space 4 over denominator 9 end fraction left parenthesis x space plus space a right parenthesis squared space space space space space space space space space space space space space space space space
space space space space space space space space space space space space space space space space space space space space space space p over q space equals space 4 over 9 space cross times space 2 over 3
space space space space space space space space space space space space space space space space space space space space space p over q space equals 8 over 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)
For such questions, we should know the formulas of subnormal and subtangent.
Related Questions to study
If the radius of earth is R then height 'h ' at which value of ' g' becomes one - fourth is
If the radius of earth is R then height 'h ' at which value of ' g' becomes one - fourth is
An object weights 72 N on the earth. Its weight at a height R/2 from earth is ……N
An object weights 72 N on the earth. Its weight at a height R/2 from earth is ……N
If mass of a body is M on the earth surface, than the mass of the same body on the moont surfae is
If mass of a body is M on the earth surface, than the mass of the same body on the moont surfae is
If the density of small planet is that of the same as that of the earth while the radius of the planet is 0.2 times that of lhe earth, the gravitational acceleration on the surface of the planet is……
If the density of small planet is that of the same as that of the earth while the radius of the planet is 0.2 times that of lhe earth, the gravitational acceleration on the surface of the planet is……
The depth of at which the value of acceleration due to gravity becomes 1/n the time the value of at the surface is (R= radius of earth)
The depth of at which the value of acceleration due to gravity becomes 1/n the time the value of at the surface is (R= radius of earth)
The moon's radius is 1/4 that of earth and its mass is 1/80 times that of the earth. If g represents the acceleration due to gravity on the surface of earth, that on the surface of the moon is
The moon's radius is 1/4 that of earth and its mass is 1/80 times that of the earth. If g represents the acceleration due to gravity on the surface of earth, that on the surface of the moon is
The length of the tangent of the curves
is
For such questions, we should know the properties of trignometric functions and their different formulas.
The length of the tangent of the curves
is
For such questions, we should know the properties of trignometric functions and their different formulas.
If earth rotates faster than its present speed the weight of an object will.
If earth rotates faster than its present speed the weight of an object will.
The value of g on be earth surface is 980 cm/
. Its value at a height of 64 km from the earth surface is....cm5-2
The value of g on be earth surface is 980 cm/
. Its value at a height of 64 km from the earth surface is....cm5-2
A body weights 700 g wt on the surface of earth How much it weight on the surface of planet whose mass is 1/7 and radius is half that of the earth
A body weights 700 g wt on the surface of earth How much it weight on the surface of planet whose mass is 1/7 and radius is half that of the earth
in a sine wave, position of different particles at time t =0 is sbown in figure: The equation for this wave travelling along the positive x - direction can be
![](data:image/png;base64,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)
in a sine wave, position of different particles at time t =0 is sbown in figure: The equation for this wave travelling along the positive x - direction can be
![](data:image/png;base64,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)
Aspherical planet far out in space has mas Mo and diameter Do. A particle of
falling near the surface of this planet will experience an acceleration due to gravity which is equal to
Aspherical planet far out in space has mas Mo and diameter Do. A particle of
falling near the surface of this planet will experience an acceleration due to gravity which is equal to
If
then ![f subscript x y end subscript equals](data:image/png;base64,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)
We have used rules of indices and different formulae of differentiation to solve the question. When we find the derivative w.r.t to some variable we take another variable as consta.
If
then ![f subscript x y end subscript equals](data:image/png;base64,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)
We have used rules of indices and different formulae of differentiation to solve the question. When we find the derivative w.r.t to some variable we take another variable as consta.