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The normal at any point q to the curve x = a (cos q + q sin q), y = a (sin q – q cos q) is at distance from the origin that is equal to… .
- a)a
- b)2a
- c)4a
- d)3a
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The normal at any point q to the curve x = a (cos q + q sin q), y = a ...
Clearly, dx/dy =tan q = slope of normal = −cotq
Equation of normal at θ' is y−a(sinq − qcosq) = −cotq(x−a(cosq + qsinq)
= ysinq − asin2q + aqcosqsinq
= −xcosq + acos2q + aqsinqcosq
= xcosq + ysinq = a
Equation of normal at θ' is y−a(sinq − qcosq) = −cotq(x−a(cosq + qsinq)
= ysinq − asin2q + aqcosqsinq
= −xcosq + acos2q + aqsinqcosq
= xcosq + ysinq = a
Most Upvoted Answer
The normal at any point q to the curve x = a (cos q + q sin q), y = a ...
The normal at any point q to the curve x = a (cos q), y = a (sin q) can be found by taking the derivative of y with respect to x, and then taking the negative reciprocal of that derivative.
First, we find the derivative of y with respect to x using the chain rule:
dy/dx = (dy/dq)/(dx/dq)
Since x = a (cos q) and y = a (sin q), we can find dx/dq and dy/dq:
dx/dq = -a(sin q)
dy/dq = a(cos q)
Plugging these values into the derivative expression:
dy/dx = (a(cos q))/(-a(sin q))
= -cos q / sin q
= -cot q
To find the normal, we take the negative reciprocal of dy/dx:
dy/dx = -cot q
Normal: dx/dy = -1 / dy/dx = -1 / (-cot q) = tan q
Therefore, the normal at any point q to the curve x = a (cos q), y = a (sin q) is given by the equation y = tan q.
First, we find the derivative of y with respect to x using the chain rule:
dy/dx = (dy/dq)/(dx/dq)
Since x = a (cos q) and y = a (sin q), we can find dx/dq and dy/dq:
dx/dq = -a(sin q)
dy/dq = a(cos q)
Plugging these values into the derivative expression:
dy/dx = (a(cos q))/(-a(sin q))
= -cos q / sin q
= -cot q
To find the normal, we take the negative reciprocal of dy/dx:
dy/dx = -cot q
Normal: dx/dy = -1 / dy/dx = -1 / (-cot q) = tan q
Therefore, the normal at any point q to the curve x = a (cos q), y = a (sin q) is given by the equation y = tan q.
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