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The locus of the mid point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix
- a)x = -a
- b)x = -a/2
- c)x = 0
- d)x = a/2
Correct answer is option 'C'. Can you explain this answer?
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The locus of the mid point of the line segment joining the focus to a ...
Locus of the midpoint of the line segment joining the focus to a moving point on the parabola y^2 = 4ax
The given equation represents a parabola with its vertex at the origin (0, 0) and its axis of symmetry along the x-axis. The parameter 'a' determines the shape and orientation of the parabola.
To find the locus of the midpoint, we need to consider a point P on the parabola with coordinates (x, y). The coordinates of the focus F can be found using the relation F = (a/4, 0).
Let M be the midpoint of the line segment joining F and P, with coordinates (h, k).
Finding the coordinates of the midpoint:
Since M is the midpoint of the line segment joining F and P, we can use the midpoint formula to find its coordinates:
h = (x + a/4)/2
k = y/2
Simplifying the equations:
2h = x + a/4
2k = y
Substituting the value of y from the given equation of the parabola:
2k = 4ax
Simplifying further:
k = 2ax
Therefore, the coordinates of the midpoint M are (h, k) = (x + a/4, 2ax).
Locus of the midpoint:
To find the locus of the midpoint, we need to eliminate the variable x from the coordinates of M.
From the equation h = x + a/4, we can rewrite it as x = h - a/4.
Substituting this value of x in the equation for k, we have:
k = 2a(h - a/4)
k = 2ah - a^2/2
Simplifying further:
a^2/2 - 2ah + k = 0
This is the equation of a parabola with the directrix x = -a/2 and the focus at the origin (0, 0). Therefore, the correct answer is option C: x = -a/2.
The given equation represents a parabola with its vertex at the origin (0, 0) and its axis of symmetry along the x-axis. The parameter 'a' determines the shape and orientation of the parabola.
To find the locus of the midpoint, we need to consider a point P on the parabola with coordinates (x, y). The coordinates of the focus F can be found using the relation F = (a/4, 0).
Let M be the midpoint of the line segment joining F and P, with coordinates (h, k).
Finding the coordinates of the midpoint:
Since M is the midpoint of the line segment joining F and P, we can use the midpoint formula to find its coordinates:
h = (x + a/4)/2
k = y/2
Simplifying the equations:
2h = x + a/4
2k = y
Substituting the value of y from the given equation of the parabola:
2k = 4ax
Simplifying further:
k = 2ax
Therefore, the coordinates of the midpoint M are (h, k) = (x + a/4, 2ax).
Locus of the midpoint:
To find the locus of the midpoint, we need to eliminate the variable x from the coordinates of M.
From the equation h = x + a/4, we can rewrite it as x = h - a/4.
Substituting this value of x in the equation for k, we have:
k = 2a(h - a/4)
k = 2ah - a^2/2
Simplifying further:
a^2/2 - 2ah + k = 0
This is the equation of a parabola with the directrix x = -a/2 and the focus at the origin (0, 0). Therefore, the correct answer is option C: x = -a/2.
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The locus of the mid point of the line segment joining the focus to a ...
Let p (h, k) be the mid point of the line segment joining the focus (a, 0) and a general point Q (x, y) on the parabola. Then
Put these values of x and y in y2 = 4ax, we get
Put these values of x and y in y2 = 4ax, we get
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The locus of the mid point of the line segment joining the focus to a moving point on the parabola y2= 4ax is another parabola with directrixa)x = -ab)x = -a/2c)x = 0d)x = a/2Correct answer is option 'C'. Can you explain this answer?
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