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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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Most Upvoted Answer
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,
⇒ x = 2h − a, y = 2k
Put these values of x and y in y2 = 4ax to get
4k2 = 4a(2h − a)
⇒ 4k2 = 8ah − 4a2
⇒ k2 = 2ah − a2
So, the locus of P(h, k) is y2 = 2ax − a2
Its directrix is x − a/2 = −a/2
⇒ x = 0.
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The locus of the mid point of the line segment joining the focus to a...
Introduction:
The given equation is y^2 = 4ax, which represents a parabola with vertex at the origin and the focus at (a/4, 0). We need to find the locus of the midpoint of the line segment joining the focus to a moving point on the parabola.
Understanding the problem:
To solve this problem, we need to find the equation of the locus of the midpoint of the line segment joining the focus to a moving point on the parabola. We know that the midpoint of a line segment is the average of its endpoints.
Solution:
Let P(x, y) be a point on the parabola y^2 = 4ax. The coordinates of the focus F are (a/4, 0). The midpoint M of the line segment PF can be found by taking the average of the x-coordinates and the y-coordinates.
Step 1: Finding the x-coordinate of the midpoint:
The x-coordinate of the midpoint M is the average of the x-coordinates of P and F. Therefore, the x-coordinate of M is (x + a/4)/2 = (x + a/4)/2.
Step 2: Finding the y-coordinate of the midpoint:
The y-coordinate of the midpoint M is the average of the y-coordinates of P and F. Therefore, the y-coordinate of M is (y + 0)/2 = y/2.
Step 3: Writing the equation of the locus:
Now we have the coordinates of the midpoint M as (x + a/4)/2, y/2. To find the equation of the locus, we need to eliminate the parameter x.
Step 4: Eliminating the parameter x:
From the equation of the parabola, y^2 = 4ax, we can express x in terms of y as x = y^2/4a. Substituting this value of x in the expression for the x-coordinate of M, we get x = (y^2/4a + a/4)/2 = (y^2 + a)/8a.
Step 5: Simplifying the equation:
We have x = (y^2 + a)/8a and y/2 = (y^2 + a)/8a. Multiplying both sides of the equation y/2 = (y^2 + a)/8a by 8a, we get 4ay = y^2 + a. Rearranging the terms, we get y^2 - 4ay + a = 0.
Step 6: Comparing the equation with a standard parabola equation:
The equation y^2 - 4ay + a = 0 is in the form y^2 - 4ax = 0, which represents a parabola with vertex at the origin and directrix at x = a/2. Therefore, the locus of the midpoint of the line segment joining the focus to a moving point on the parabola y^2 = 4ax is another parabola with directrix x = a/2.
Conclusion:
The correct answer is option 'C', x = 0.
The given equation is y^2 = 4ax, which represents a parabola with vertex at the origin and the focus at (a/4, 0). We need to find the locus of the midpoint of the line segment joining the focus to a moving point on the parabola.
Understanding the problem:
To solve this problem, we need to find the equation of the locus of the midpoint of the line segment joining the focus to a moving point on the parabola. We know that the midpoint of a line segment is the average of its endpoints.
Solution:
Let P(x, y) be a point on the parabola y^2 = 4ax. The coordinates of the focus F are (a/4, 0). The midpoint M of the line segment PF can be found by taking the average of the x-coordinates and the y-coordinates.
Step 1: Finding the x-coordinate of the midpoint:
The x-coordinate of the midpoint M is the average of the x-coordinates of P and F. Therefore, the x-coordinate of M is (x + a/4)/2 = (x + a/4)/2.
Step 2: Finding the y-coordinate of the midpoint:
The y-coordinate of the midpoint M is the average of the y-coordinates of P and F. Therefore, the y-coordinate of M is (y + 0)/2 = y/2.
Step 3: Writing the equation of the locus:
Now we have the coordinates of the midpoint M as (x + a/4)/2, y/2. To find the equation of the locus, we need to eliminate the parameter x.
Step 4: Eliminating the parameter x:
From the equation of the parabola, y^2 = 4ax, we can express x in terms of y as x = y^2/4a. Substituting this value of x in the expression for the x-coordinate of M, we get x = (y^2/4a + a/4)/2 = (y^2 + a)/8a.
Step 5: Simplifying the equation:
We have x = (y^2 + a)/8a and y/2 = (y^2 + a)/8a. Multiplying both sides of the equation y/2 = (y^2 + a)/8a by 8a, we get 4ay = y^2 + a. Rearranging the terms, we get y^2 - 4ay + a = 0.
Step 6: Comparing the equation with a standard parabola equation:
The equation y^2 - 4ay + a = 0 is in the form y^2 - 4ax = 0, which represents a parabola with vertex at the origin and directrix at x = a/2. Therefore, the locus of the midpoint of the line segment joining the focus to a moving point on the parabola y^2 = 4ax is another parabola with directrix x = a/2.
Conclusion:
The correct answer is option 'C', x = 0.
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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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