
A uni-modular tangent vector on the curve at is
Answer
139.8k+ views
Hint: We know that if represents a curve then the tangent vector of is represented by . From the question, we were given the equation of curve is . Now we have to calculate . From the values of we can get the vector . Now we have to substitute . This will give us the tangent vector of at . We know that the unit vector of is . Now by using this concept, we can find the uni-modular tangent vector of curve at .
Complete step-by-step solution:
Before solving the question, we should know that if represents a curve then the tangent vector of is represented by .
From the question, we were given the equation of curve is .
Let us consider
Now let us differentiate equation (1) on both sides, then we get
Now let us differentiate equation (2) on both sides, then we get
Now let us differentiate equation (3) on both sides, then we get
From equation (4), equation (5) and equation (6), we can get the tangent vector of the curve .
Now we have to find the equation of the tangent vector.
Now we have to find the equation of the tangent at .
So, now we have to substitute in equation (7), then we get
Now we have to find the unimodular vector of equation (8).
We know that the unit vector of is .
Now let us compare with . Then we get
Now from equation (9), equation (10) and equation (11) we have to calculate the value of .
Now we have to find the unit tangent vector.
From equation (9), equation (10), equation (11) and equation (12), we get
From equation we can say that uni-modular tangent vector on the curve at is .
Hence, option A is correct.
Note: Students may do the solution up to equation (8) and they may conclude that is the tangent vector. They may think that this is the correct answer but this will give us the wrong answer. The reason is that in the question it was given that to find the uni-modular vector. We know that will not represents a uni -modular vector. So, students should follow the question properly and then proceed through the solution.
Complete step-by-step solution:
Before solving the question, we should know that if
From the question, we were given the equation of curve is
Let us consider
Now let us differentiate equation (1) on both sides, then we get
Now let us differentiate equation (2) on both sides, then we get
Now let us differentiate equation (3) on both sides, then we get
From equation (4), equation (5) and equation (6), we can get the tangent vector of the curve
Now we have to find the equation of the tangent vector.
Now we have to find the equation of the tangent at
So, now we have to substitute
Now we have to find the unimodular vector of equation (8).
We know that the unit vector of
Now let us compare
Now from equation (9), equation (10) and equation (11) we have to calculate the value of
Now we have to find the unit tangent vector.
From equation (9), equation (10), equation (11) and equation (12), we get
From equation we can say that uni-modular tangent vector on the curve
Hence, option A is correct.
Note: Students may do the solution up to equation (8) and they may conclude that
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