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Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 intersect the y-axis at (0, 3/2). If m is the slope of the tangent at P to the curve, then |m| is equal to _______.
Correct answer is '4'. Can you explain this answer?
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Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 inters...
Given curve: y^2 - 3x^2 - y - 10 = 0
To find the slope of the tangent at point P on the curve, we need to find the value of m.
Finding the coordinates of point P:
Let the coordinates of point P be (x, y).
Substituting the values of x and y in the equation of the curve:
y^2 - 3x^2 - y - 10 = 0
We can rearrange the equation as:
y^2 - y - 3x^2 - 10 = 0
Since P lies on the curve, this equation must hold true.
Finding the coordinates of the point of intersection with the y-axis:
The normal at point P intersects the y-axis at the point (0, 3/2).
Substituting x = 0 and y = 3/2 in the equation of the curve:
(3/2)^2 - 3(0)^2 - (3/2) - 10 = 0
9/4 - 3/2 - 3/2 - 10 = 0
9/4 - 3 - 10 = 0
9/4 - 13/4 = 0
-4/4 = 0
0 = 0
Since this equation holds true, the point (0, 3/2) lies on the curve.
Finding the slope of the normal at point P:
The slope of the normal at point P is given by the negative reciprocal of the derivative of the curve at point P.
Differentiating the equation of the curve with respect to x:
d/dx(y^2 - 3x^2 - y - 10) = 0
2yy' - 6x - y' = 0
2yy' - y' = 6x
y'(2y - 1) = 6x
y' = 6x / (2y - 1)
Substituting the coordinates of point P in the above equation:
m = 6x / (2y - 1)
m = 6x / (2(y)) - 1)
m = 6x / (2y - 1)
Since the normal intersects the y-axis at (0, 3/2), we can substitute x = 0 and y = 3/2 in the above equation:
m = 6(0) / (2(3/2) - 1)
m = 0 / (3 - 1)
m = 0 / 2
m = 0
Thus, the slope of the tangent at point P is 0.
Since the question asks for the absolute value of m, we have |m| = |0| = 0.
Therefore, the correct answer is 0.
To find the slope of the tangent at point P on the curve, we need to find the value of m.
Finding the coordinates of point P:
Let the coordinates of point P be (x, y).
Substituting the values of x and y in the equation of the curve:
y^2 - 3x^2 - y - 10 = 0
We can rearrange the equation as:
y^2 - y - 3x^2 - 10 = 0
Since P lies on the curve, this equation must hold true.
Finding the coordinates of the point of intersection with the y-axis:
The normal at point P intersects the y-axis at the point (0, 3/2).
Substituting x = 0 and y = 3/2 in the equation of the curve:
(3/2)^2 - 3(0)^2 - (3/2) - 10 = 0
9/4 - 3/2 - 3/2 - 10 = 0
9/4 - 3 - 10 = 0
9/4 - 13/4 = 0
-4/4 = 0
0 = 0
Since this equation holds true, the point (0, 3/2) lies on the curve.
Finding the slope of the normal at point P:
The slope of the normal at point P is given by the negative reciprocal of the derivative of the curve at point P.
Differentiating the equation of the curve with respect to x:
d/dx(y^2 - 3x^2 - y - 10) = 0
2yy' - 6x - y' = 0
2yy' - y' = 6x
y'(2y - 1) = 6x
y' = 6x / (2y - 1)
Substituting the coordinates of point P in the above equation:
m = 6x / (2y - 1)
m = 6x / (2(y)) - 1)
m = 6x / (2y - 1)
Since the normal intersects the y-axis at (0, 3/2), we can substitute x = 0 and y = 3/2 in the above equation:
m = 6(0) / (2(3/2) - 1)
m = 0 / (3 - 1)
m = 0 / 2
m = 0
Thus, the slope of the tangent at point P is 0.
Since the question asks for the absolute value of m, we have |m| = |0| = 0.
Therefore, the correct answer is 0.
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Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 inters...
y2 - 3x2 + y + 10 = 0
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Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 intersect the y-axis at (0, 3/2).If m is the slope of the tangent at P to the curve, then |m| is equal to _______.Correct answer is '4'. Can you explain this answer?
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Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 intersect the y-axis at (0, 3/2).If m is the slope of the tangent at P to the curve, then |m| is equal to _______.Correct answer is '4'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 intersect the y-axis at (0, 3/2).If m is the slope of the tangent at P to the curve, then |m| is equal to _______.Correct answer is '4'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 intersect the y-axis at (0, 3/2).If m is the slope of the tangent at P to the curve, then |m| is equal to _______.Correct answer is '4'. Can you explain this answer?.
Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 intersect the y-axis at (0, 3/2).If m is the slope of the tangent at P to the curve, then |m| is equal to _______.Correct answer is '4'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 intersect the y-axis at (0, 3/2).If m is the slope of the tangent at P to the curve, then |m| is equal to _______.Correct answer is '4'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let the normal at a point P on the curves y2 - 3x2 + y + 10 = 0 intersect the y-axis at (0, 3/2).If m is the slope of the tangent at P to the curve, then |m| is equal to _______.Correct answer is '4'. Can you explain this answer?.
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