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The co-ordinates of the focus of the parabola described parametrically by x = 5t2 + 2, y = 10t + 4 are
- a)(7, 4)
- b)(3, 4)
- c)(3, –4)
- d)(–7, 4)
Correct answer is option 'A'. Can you explain this answer?
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The co-ordinates of the focus of the parabola described parametrically...
x = 5t2 + 2 ; y = 10t + 4 , 
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The co-ordinates of the focus of the parabola described parametrically...
The standard form of a parabola with focus at (a, b) and directrix y = k is given by:
(x - a)^2 + (y - b)^2 = (y - k)^2
In this case, we can rewrite the parametric equations as:
x = 5t^2 + 2
y = 10t + 4
Solving for t in terms of y, we get:
t = (y - 4) / 10
Substituting this expression for t into the equation for x, we get:
x = 5[(y - 4) / 10]^2 + 2
x = (y - 4)^2 / 20 + 2
Comparing this to the standard form, we see that the focus is at (2, 4 + p), where p is the distance from the focus to the directrix. To find p, we can use the fact that the distance from a point (x, y) to the directrix y = k is given by |y - k|.
Since the directrix in this case is y = 4, we have:
p = |(y - 4) - 10t| / 2
p = |y - 10t - 4| / 2
p = |y - x/5 - 4| / 2
Substituting our expressions for x and y, we get:
p = |(y - 4)^2 / 20 - y/5| / 2
p = |(y^2 - 8y + 16 - 4y) / 100| / 2
p = |(y^2 - 12y + 16) / 100| / 2
p = |y^2 - 12y + 16| / 200
To minimize calculation, we can recognize that the vertex of the parabola is at (2, 4) and the axis of symmetry is vertical. Therefore, the focus must also lie on this axis, and its y-coordinate is given by 4 + p.
We can now plug in the answer choices and see which one gives us the correct value for p:
a) p = |7^2 - 12(7) + 16| / 200 = 3 / 25
b) p = |3^2 - 12(3) + 16| / 200 = 1 / 25
c) p = |3^2 - 12(3) + 25| / 200 = 1 / 20
Therefore, the answer is (c) (3, 4 + 1/20) = (3, 4.05).
(x - a)^2 + (y - b)^2 = (y - k)^2
In this case, we can rewrite the parametric equations as:
x = 5t^2 + 2
y = 10t + 4
Solving for t in terms of y, we get:
t = (y - 4) / 10
Substituting this expression for t into the equation for x, we get:
x = 5[(y - 4) / 10]^2 + 2
x = (y - 4)^2 / 20 + 2
Comparing this to the standard form, we see that the focus is at (2, 4 + p), where p is the distance from the focus to the directrix. To find p, we can use the fact that the distance from a point (x, y) to the directrix y = k is given by |y - k|.
Since the directrix in this case is y = 4, we have:
p = |(y - 4) - 10t| / 2
p = |y - 10t - 4| / 2
p = |y - x/5 - 4| / 2
Substituting our expressions for x and y, we get:
p = |(y - 4)^2 / 20 - y/5| / 2
p = |(y^2 - 8y + 16 - 4y) / 100| / 2
p = |(y^2 - 12y + 16) / 100| / 2
p = |y^2 - 12y + 16| / 200
To minimize calculation, we can recognize that the vertex of the parabola is at (2, 4) and the axis of symmetry is vertical. Therefore, the focus must also lie on this axis, and its y-coordinate is given by 4 + p.
We can now plug in the answer choices and see which one gives us the correct value for p:
a) p = |7^2 - 12(7) + 16| / 200 = 3 / 25
b) p = |3^2 - 12(3) + 16| / 200 = 1 / 25
c) p = |3^2 - 12(3) + 25| / 200 = 1 / 20
Therefore, the answer is (c) (3, 4 + 1/20) = (3, 4.05).
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The co-ordinates of the focus of the parabola described parametrically by x = 5t2 + 2, y = 10t + 4 area)(7, 4)b)(3, 4)c)(3, –4)d)(–7, 4)Correct answer is option 'A'. Can you explain this answer?
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