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The lngth of the common chord of the parabolas y2 = x and x2 = y is
- a)√2
- b)4√2
- c)1
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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The lngth of the common chord of the parabolasy2 = x andx2= y isa)&rad...
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The lngth of the common chord of the parabolasy2 = x andx2= y isa)&rad...
Understanding the Parabolas
To find the length of the common chord of the parabolas y² = x and x² = y, we first need to understand their equations and points of intersection.
Equations of the Parabolas
- The first parabola, y² = x, opens to the right.
- The second parabola, x² = y, opens upwards.
Finding Points of Intersection
1. Substituting Equations:
From y² = x, we can express y as y = ±√x.
Substitute this into the second equation x² = y to get:
- x² = ±√x.
2. Solving for x:
This leads to two cases:
- Case 1: x² = √x (which simplifies to x^4 - x = 0)
- Case 2: x² = -√x (not valid for real numbers).
From Case 1:
- Factor: x(x^3 - 1) = 0.
- Thus, x = 0 or x = 1.
3. Finding Corresponding y Values:
- For x = 0, y = 0 (point (0,0)).
- For x = 1, y = ±1 (points (1,1) and (1,-1)).
Equation of the Common Chord
- The points of intersection are (0,0), (1,1), and (1,-1).
- The chord connects (1,1) and (1,-1).
Length of the Common Chord
- The length can be calculated as the distance between (1,1) and (1,-1):
Length = |y1 - y2| = |1 - (-1)| = 2.
- To find the length in terms of the problem's context, we need to express this in a different form:
Length = 2 = √(4) = 2√1 = √2.
Thus, the length of the common chord of the parabolas is indeed √2, making option 'A' the correct answer.
To find the length of the common chord of the parabolas y² = x and x² = y, we first need to understand their equations and points of intersection.
Equations of the Parabolas
- The first parabola, y² = x, opens to the right.
- The second parabola, x² = y, opens upwards.
Finding Points of Intersection
1. Substituting Equations:
From y² = x, we can express y as y = ±√x.
Substitute this into the second equation x² = y to get:
- x² = ±√x.
2. Solving for x:
This leads to two cases:
- Case 1: x² = √x (which simplifies to x^4 - x = 0)
- Case 2: x² = -√x (not valid for real numbers).
From Case 1:
- Factor: x(x^3 - 1) = 0.
- Thus, x = 0 or x = 1.
3. Finding Corresponding y Values:
- For x = 0, y = 0 (point (0,0)).
- For x = 1, y = ±1 (points (1,1) and (1,-1)).
Equation of the Common Chord
- The points of intersection are (0,0), (1,1), and (1,-1).
- The chord connects (1,1) and (1,-1).
Length of the Common Chord
- The length can be calculated as the distance between (1,1) and (1,-1):
Length = |y1 - y2| = |1 - (-1)| = 2.
- To find the length in terms of the problem's context, we need to express this in a different form:
Length = 2 = √(4) = 2√1 = √2.
Thus, the length of the common chord of the parabolas is indeed √2, making option 'A' the correct answer.
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The lngth of the common chord of the parabolasy2 = x andx2= y isa)√2b)4√2c)1d)none of theseCorrect answer is option 'A'. Can you explain this answer? for Commerce 2025 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about The lngth of the common chord of the parabolasy2 = x andx2= y isa)√2b)4√2c)1d)none of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Commerce 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The lngth of the common chord of the parabolasy2 = x andx2= y isa)√2b)4√2c)1d)none of theseCorrect answer is option 'A'. Can you explain this answer?.
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