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The curve a mongst the family of curves, represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1,1) is :
- a)A circle with centre on the y-axis
- b)A circle with centre on the x-axis
- c)An ellipse with major axis along the y-axis
- d)A hyperbola with transverse axis along the x-axis
Correct answer is option 'B'. Can you explain this answer?
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The curve a mongst the family of curves, represented by the differenti...
(x^2−y^2)dx + 2xy dy=0
or,
(x^2−y^2)dx + 2xy dy=0
or,
dy/dx = (y^2−x^2)/2xy
Put :-
y=vx⇒
dy/dx=
v + x dv/dx
Solving we get,
∫2v dv/(v^2 + 1) =∫−dx/x
or,
ln(v^2 + 1) =−ln|x| +C
(y^2+x^2) = Cx
At point (1 , 1) :-
1+1=C
⇒C=2
y^2 + x^2=2x
(X - 1)^2 + y^2 = 1
center = (1 , 0)
radius = 1
THUS WE CAN SAY THAT THE OPTION (B) IS CORRECT
or,
(x^2−y^2)dx + 2xy dy=0
or,
dy/dx = (y^2−x^2)/2xy
Put :-
y=vx⇒
dy/dx=
v + x dv/dx
Solving we get,
∫2v dv/(v^2 + 1) =∫−dx/x
or,
ln(v^2 + 1) =−ln|x| +C
(y^2+x^2) = Cx
At point (1 , 1) :-
1+1=C
⇒C=2
y^2 + x^2=2x
(X - 1)^2 + y^2 = 1
center = (1 , 0)
radius = 1
THUS WE CAN SAY THAT THE OPTION (B) IS CORRECT
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The curve a mongst the family of curves, represented by the differenti...
Understanding the Differential Equation
The given differential equation is (x² - y²)dx + 2xy dy = 0. This can be rewritten in the standard form:
- dy/dx = -(x² - y²)/(2xy)
This form helps us analyze the behavior of the curves represented by the equation.
Identifying the Type of Curve
To find the specific curve that passes through the point (1, 1), we can separate variables or use an appropriate substitution. However, we can also recognize the structure of the equation.
- The equation can be rearranged and interpreted as a conic section, specifically relating to hyperbolas and circles.
Curve Through the Point (1, 1)
Substituting (x, y) = (1, 1) into the equation yields:
- 1² - 1² = 0, which indicates the solution must satisfy the curve's characteristics defined by the equation.
Determining the Correct Option
By analyzing the shape and orientation of the curves derived from the equation:
- The presence of x² and y² terms indicates that the curve could represent a conic section.
- The specific nature of the coefficients suggests that the resulting curve is likely to be a circle.
Since the curve passes through (1, 1) and is symmetric, it indicates a circle centered on the x-axis.
Conclusion
Thus, the correct answer is option 'B': "A circle with center on the x-axis." This solution aligns with the nature of conic sections derived from the given differential equation, confirming the circular shape passing through the specified point.
The given differential equation is (x² - y²)dx + 2xy dy = 0. This can be rewritten in the standard form:
- dy/dx = -(x² - y²)/(2xy)
This form helps us analyze the behavior of the curves represented by the equation.
Identifying the Type of Curve
To find the specific curve that passes through the point (1, 1), we can separate variables or use an appropriate substitution. However, we can also recognize the structure of the equation.
- The equation can be rearranged and interpreted as a conic section, specifically relating to hyperbolas and circles.
Curve Through the Point (1, 1)
Substituting (x, y) = (1, 1) into the equation yields:
- 1² - 1² = 0, which indicates the solution must satisfy the curve's characteristics defined by the equation.
Determining the Correct Option
By analyzing the shape and orientation of the curves derived from the equation:
- The presence of x² and y² terms indicates that the curve could represent a conic section.
- The specific nature of the coefficients suggests that the resulting curve is likely to be a circle.
Since the curve passes through (1, 1) and is symmetric, it indicates a circle centered on the x-axis.
Conclusion
Thus, the correct answer is option 'B': "A circle with center on the x-axis." This solution aligns with the nature of conic sections derived from the given differential equation, confirming the circular shape passing through the specified point.
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The curve a mongst the family of curves, represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1,1) is :a)A circle with centre on the y-axisb)A circle with centre on the x-axisc)An ellipse with major axis along the y-axisd)A hyperbola with transverse axis along the x-axisCorrect answer is option 'B'. Can you explain this answer?
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The curve a mongst the family of curves, represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1,1) is :a)A circle with centre on the y-axisb)A circle with centre on the x-axisc)An ellipse with major axis along the y-axisd)A hyperbola with transverse axis along the x-axisCorrect answer is option 'B'. 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 The curve a mongst the family of curves, represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1,1) is :a)A circle with centre on the y-axisb)A circle with centre on the x-axisc)An ellipse with major axis along the y-axisd)A hyperbola with transverse axis along the x-axisCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The curve a mongst the family of curves, represented by the differential equation, (x2 – y2)dx + 2xy dy = 0 which passes through (1,1) is :a)A circle with centre on the y-axisb)A circle with centre on the x-axisc)An ellipse with major axis along the y-axisd)A hyperbola with transverse axis along the x-axisCorrect answer is option 'B'. Can you explain this answer?.
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