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Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of
- a)circles
- b)parabolas
- c)hyperbolas
- d)straight lines
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Through any point (x, y) of a curve which passes through the origin, l...
Let P(x, y) be the point on the curve passing through the origin O(0, 0), and let PN and PM be the lines parallel to the x- and y-axes, respectively. If the equation of the curve is y = y(x), the area POM equals
and the 
This solution represents a parabola. We will get a similar result if we had started instead with 2(PON) = POM.
Most Upvoted Answer
Through any point (x, y) of a curve which passes through the origin, l...
Explanation:
We are given that a curve passes through the origin and divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other. Let us assume that the equation of the curve is y = f(x).
Area of the rectangle: The area of the rectangle formed by the two lines and the axes is given by the product of the lengths of the two sides. Let the length of the x-axis be a and the length of the y-axis be b. Then, the area of the rectangle is ab.
Area of the regions: The region above the curve and below the x-axis is the region A1. The region below the curve and above the x-axis is the region A2. These two regions divide the rectangle into two parts.
Equation of the curve: Let us assume that the equation of the curve passing through the origin is y = f(x).
Region A1: The area of region A1 is given by the integral of f(x) from 0 to x. This is because the curve passes through the origin and the region A1 is above the curve and below the x-axis.
Region A2: The area of region A2 is given by the integral of -f(x) from 0 to x. This is because the curve passes through the origin and the region A2 is below the curve and above the x-axis.
Equation of the curve: The equation of the curve can be found by equating the two areas. That is, A1 = 2A2. This gives us the equation:
∫0x f(x) dx = 2∫0x -f(x) dx
Simplifying this equation, we get:
∫0x 3f(x) dx = 0
Since this equation holds for all values of x, we can conclude that 3f(x) = 0, which implies that f(x) = 0.
Therefore, the equation of the curve passing through the origin and dividing the rectangle into two parts, one of which is twice the other, is y = 0, which is the equation of the x-axis.
Hence, the correct answer is option 'B', parabolas, which is not a valid option. Therefore, the answer to the question is that none of the given options is correct.
We are given that a curve passes through the origin and divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other. Let us assume that the equation of the curve is y = f(x).
Area of the rectangle: The area of the rectangle formed by the two lines and the axes is given by the product of the lengths of the two sides. Let the length of the x-axis be a and the length of the y-axis be b. Then, the area of the rectangle is ab.
Area of the regions: The region above the curve and below the x-axis is the region A1. The region below the curve and above the x-axis is the region A2. These two regions divide the rectangle into two parts.
Equation of the curve: Let us assume that the equation of the curve passing through the origin is y = f(x).
Region A1: The area of region A1 is given by the integral of f(x) from 0 to x. This is because the curve passes through the origin and the region A1 is above the curve and below the x-axis.
Region A2: The area of region A2 is given by the integral of -f(x) from 0 to x. This is because the curve passes through the origin and the region A2 is below the curve and above the x-axis.
Equation of the curve: The equation of the curve can be found by equating the two areas. That is, A1 = 2A2. This gives us the equation:
∫0x f(x) dx = 2∫0x -f(x) dx
Simplifying this equation, we get:
∫0x 3f(x) dx = 0
Since this equation holds for all values of x, we can conclude that 3f(x) = 0, which implies that f(x) = 0.
Therefore, the equation of the curve passing through the origin and dividing the rectangle into two parts, one of which is twice the other, is y = 0, which is the equation of the x-axis.
Hence, the correct answer is option 'B', parabolas, which is not a valid option. Therefore, the answer to the question is that none of the given options is correct.
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Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of a)circles b)parabolas c)hyperbolas d)straight lines Correct answer is option 'B'. Can you explain this answer?
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Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of a)circles b)parabolas c)hyperbolas d)straight lines Correct answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of a)circles b)parabolas c)hyperbolas d)straight lines Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of a)circles b)parabolas c)hyperbolas d)straight lines Correct answer is option 'B'. Can you explain this answer?.
Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of a)circles b)parabolas c)hyperbolas d)straight lines Correct answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of a)circles b)parabolas c)hyperbolas d)straight lines Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Through any point (x, y) of a curve which passes through the origin, lines are drawn parallel to the coordinate axes. The curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of a)circles b)parabolas c)hyperbolas d)straight lines Correct answer is option 'B'. Can you explain this answer?.
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