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Angle of intersection between the curves y2=4ax and x2=4ay?
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Angle of intersection between the curves y2=4ax and x2=4ay?
Angle of Intersection between Curves y^2=4ax and x^2=4ay
When two curves intersect, the angle at the point of intersection is the angle between the tangents at that point. Therefore, to find the angle of intersection between the curves y^2=4ax and x^2=4ay, we need to find the equations of their tangents at the point of intersection.
Step 1: Find the Point of Intersection
To find the point of intersection between the two curves, we can substitute y^2=4ax into x^2=4ay:
x^2=4ay
x^2=4a(sqrt(x))
x^(3/2)=4a
x=(4a)^(2/3)
Substituting this value of x into y^2=4ax:
y^2=4a(4a)^(2/3)
y=2(4a)^(1/3)
Therefore, the point of intersection between the two curves is (x,y) = ((4a)^(2/3), 2(4a)^(1/3)).
Step 2: Find the Equations of the Tangents
To find the equations of the tangents at the point of intersection, we need to find the derivatives of the two curves at that point:
y^2=4ax
2yy' = 4a
y' = 2a/y
x^2=4ay
2xx' = 4a
x' = 2a/x
Substituting the values of x and y at the point of intersection, we get:
y' = 2a/2(4a)^(1/3) = (4a)^(1/3)/a
x' = 2a/(4a)^(2/3) = (2a)^(1/3)/(4a)^(1/3)
Therefore, the equations of the tangents at the point of intersection are:
y = (4a)^(1/3)/a(x-(4a)^(2/3)) + 2(4a)^(1/3)
y = (2a)^(1/3)/(4a)^(1/3)(x-(4a)^(2/3)) + 2(4a)^(1/3)
Step 3: Find the Angle of Intersection
The angle of intersection between the two curves is the angle between the two tangents at the point of intersection. We can find this angle using the formula:
tan(theta) = |(m2-m1)/(1+m1m2)|
where m1 and m2 are the slopes of the two tangents.
Substituting the values of m1 and m2, we get:
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Angle of intersection between the curves y2=4ax and x2=4ay?
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Angle of intersection between the curves y2=4ax and x2=4ay?
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Angle of intersection between the curves y2=4ax and x2=4ay? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Angle of intersection between the curves y2=4ax and x2=4ay? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Angle of intersection between the curves y2=4ax and x2=4ay?.
Angle of intersection between the curves y2=4ax and x2=4ay? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Angle of intersection between the curves y2=4ax and x2=4ay? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Angle of intersection between the curves y2=4ax and x2=4ay?.
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