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The locus of the mid point of the focal radii of a variable point moving on the parabola, y2 = 4ax is a parabola whose
- a)latus rectum is half the latus rectum of the original parabola
- b)vertex is (a/2, 0)
- c)directrix is y-axis
- d)focus has the co-ordinates (a, 0)
Correct answer is option 'A,B,C,D'. Can you explain this answer?
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The locus of the mid point of the focal radii of a variable point movi...
Understanding the Problem
In this problem, we need to find the locus of the midpoint of the focal radii of a point moving on the parabola y² = 4ax.
Locus of the Midpoint
1. A point (x, y) on the parabola can be represented as (at², 2at).
2. The focus of the parabola is (a, 0), and the directrix is the line x = -a.
3. The distances from the point (at², 2at) to the focus and directrix must be calculated to find the focal radii.
Finding the Midpoint
1. The focal radius to the focus (a, 0) is given by the distance formula.
2. The midpoint of the focal radii can be expressed as:
- Midpoint (M) = ((at² + a)/2, (2at + 0)/2) = ((at² + a)/2, at).
Equation of the New Parabola
1. To derive the equation of the locus of point M:
- Let x = (at² + a)/2 => at² = 2x - a.
- Substitute at² into y = at to find the relationship between x and y.
2. This results in a new parabola equation that can be simplified to show that it retains the parabolic form.
Properties of the New Parabola
- The new parabola has:
- a latus rectum that is half that of the original parabola.
- a vertex at (a/2, 0).
- a directrix along the y-axis.
- a focus located at (a, 0).
Conclusion
Thus, the locus of the midpoint of the focal radii forms a parabola with the specified properties:
- Latus Rectum: Half of the original parabola's latus rectum.
- Vertex: (a/2, 0).
- Directrix: y-axis.
- Focus: (a, 0).
Therefore, the correct answer encompasses all options: A, B, C, D.
In this problem, we need to find the locus of the midpoint of the focal radii of a point moving on the parabola y² = 4ax.
Locus of the Midpoint
1. A point (x, y) on the parabola can be represented as (at², 2at).
2. The focus of the parabola is (a, 0), and the directrix is the line x = -a.
3. The distances from the point (at², 2at) to the focus and directrix must be calculated to find the focal radii.
Finding the Midpoint
1. The focal radius to the focus (a, 0) is given by the distance formula.
2. The midpoint of the focal radii can be expressed as:
- Midpoint (M) = ((at² + a)/2, (2at + 0)/2) = ((at² + a)/2, at).
Equation of the New Parabola
1. To derive the equation of the locus of point M:
- Let x = (at² + a)/2 => at² = 2x - a.
- Substitute at² into y = at to find the relationship between x and y.
2. This results in a new parabola equation that can be simplified to show that it retains the parabolic form.
Properties of the New Parabola
- The new parabola has:
- a latus rectum that is half that of the original parabola.
- a vertex at (a/2, 0).
- a directrix along the y-axis.
- a focus located at (a, 0).
Conclusion
Thus, the locus of the midpoint of the focal radii forms a parabola with the specified properties:
- Latus Rectum: Half of the original parabola's latus rectum.
- Vertex: (a/2, 0).
- Directrix: y-axis.
- Focus: (a, 0).
Therefore, the correct answer encompasses all options: A, B, C, D.
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The locus of the mid point of the focal radii of a variable point moving on the parabola, y2= 4ax is a parabola whosea)latus rectum is half the latus rectum of the original parabolab)vertex is (a/2, 0)c)directrix is y-axisd)focus has the co-ordinates (a, 0)Correct answer is option 'A,B,C,D'. Can you explain this answer?
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