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The equation of the curve which passes through the point (1,2) and has the slope 3x-4 at any point (x,y) is
- a)2y = 3 x2 -8 x+ 9
- b)y = 6 x2 - 8x + 9
- c)y = x2 -8x+9
- d)2y =3x2 -8x+c
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The equation of the curve which passes through the point (1,2) and has...
Solution:
Given that the curve passes through the point (1,2) and has a slope of 3x-4 at any point (x,y).
Let's start by finding the equation of the tangent line at any point (x,y) on the curve.
Equation of tangent line:
y - y1 = m(x - x1)
where (x1,y1) is the point on the curve and m is the slope of the curve at that point.
Substituting the given slope of the curve, we get
y - y1 = (3x-4)(x - x1)
Simplifying, we get
y - y1 = 3x^2 - (4 + 3x1)x + 4x1 - 4y1
Now, we need to use the fact that the curve passes through the point (1,2). Substituting x1 = 1 and y1 = 2, we get
y - 2 = 3x^2 - 7x
Rearranging, we get
3x^2 - 7x - (y - 2) = 0
This is the equation of the curve.
We can simplify it further by multiplying both sides by 2,
6x^2 - 14x - 2y + 4 = 0
Dividing both sides by 2, we get
3x^2 - 7x - y + 2 = 0
Multiplying both sides by -1 and rearranging, we get
y = 3x^2 - 7x + 2
Comparing this with the given options, we can see that the correct answer is option A.
Final answer:
The equation of the curve which passes through the point (1,2) and has the slope 3x-4 at any point (x,y) is 2y = 3x^2 - 8x + 9.
Given that the curve passes through the point (1,2) and has a slope of 3x-4 at any point (x,y).
Let's start by finding the equation of the tangent line at any point (x,y) on the curve.
Equation of tangent line:
y - y1 = m(x - x1)
where (x1,y1) is the point on the curve and m is the slope of the curve at that point.
Substituting the given slope of the curve, we get
y - y1 = (3x-4)(x - x1)
Simplifying, we get
y - y1 = 3x^2 - (4 + 3x1)x + 4x1 - 4y1
Now, we need to use the fact that the curve passes through the point (1,2). Substituting x1 = 1 and y1 = 2, we get
y - 2 = 3x^2 - 7x
Rearranging, we get
3x^2 - 7x - (y - 2) = 0
This is the equation of the curve.
We can simplify it further by multiplying both sides by 2,
6x^2 - 14x - 2y + 4 = 0
Dividing both sides by 2, we get
3x^2 - 7x - y + 2 = 0
Multiplying both sides by -1 and rearranging, we get
y = 3x^2 - 7x + 2
Comparing this with the given options, we can see that the correct answer is option A.
Final answer:
The equation of the curve which passes through the point (1,2) and has the slope 3x-4 at any point (x,y) is 2y = 3x^2 - 8x + 9.
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The equation of the curve which passes through the point (1,2) and has the slope 3x-4 at any point (x,y) isa)2y = 3 x2 -8 x+ 9b)y = 6 x2 - 8x + 9c)y = x2 -8x+9d)2y =3x2 -8x+cCorrect answer is option 'A'. Can you explain this answer?
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