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The vertex of a parabola is (1,2) and its axis is parallel to y-axis. If parabola passes through (0,6), then the length of its latus rectum is _____
- a)1/2
- b)2
- c)1/4
- d)4
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
The vertex of a parabola is (1,2) and its axis is parallel to y-axis....
To find the length of the latus rectum of the given parabola, we need to determine the equation of the parabola first. Given that the vertex is (1,2) and the axis is parallel to the y-axis, we can write the equation of the parabola in the following form:
(x - h)^2 = 4a(y - k)
where (h, k) is the vertex of the parabola. Substituting the given values, we have:
(x - 1)^2 = 4a(y - 2)
Since the parabola passes through the point (0,6), we can substitute these coordinates into the equation:
(0 - 1)^2 = 4a(6 - 2)
Simplifying the equation, we get:
1 = 16a
Therefore, the value of 'a' is 1/16.
Now, we know the value of 'a', we can find the equation of the parabola:
(x - 1)^2 = 4(1/16)(y - 2)
Simplifying further:
(x - 1)^2 = (1/4)(y - 2)
Equation of the parabola is obtained.
Next, we know that the latus rectum of a parabola is given by the equation:
4a = length of latus rectum
Substituting the value of 'a' as 1/16, we have:
4(1/16) = length of latus rectum
Simplifying, we find:
1/4 = length of latus rectum
Therefore, the length of the latus rectum of the given parabola is 1/4.
Hence, option 'C' is the correct answer.
(x - h)^2 = 4a(y - k)
where (h, k) is the vertex of the parabola. Substituting the given values, we have:
(x - 1)^2 = 4a(y - 2)
Since the parabola passes through the point (0,6), we can substitute these coordinates into the equation:
(0 - 1)^2 = 4a(6 - 2)
Simplifying the equation, we get:
1 = 16a
Therefore, the value of 'a' is 1/16.
Now, we know the value of 'a', we can find the equation of the parabola:
(x - 1)^2 = 4(1/16)(y - 2)
Simplifying further:
(x - 1)^2 = (1/4)(y - 2)
Equation of the parabola is obtained.
Next, we know that the latus rectum of a parabola is given by the equation:
4a = length of latus rectum
Substituting the value of 'a' as 1/16, we have:
4(1/16) = length of latus rectum
Simplifying, we find:
1/4 = length of latus rectum
Therefore, the length of the latus rectum of the given parabola is 1/4.
Hence, option 'C' is the correct answer.
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Community Answer
The vertex of a parabola is (1,2) and its axis is parallel to y-axis....
The equation will be of the form
(y − 2) = λ(x − 1)2
Passing through (0,6)
⇒ 4 = λ(0 − 1)2
⇒ λ = 4
⇒ (y − 2) = 4(x−1)2Comparing with x2 = 4ay, we get
The length of latus rectum = 4a = 1/4
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