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If the latus rectum of a hyperbola through one focus, subtends 60 ° angle at the other focus, then its eccentricity is
- a)√2
- b)√3
- c)√5
- d)√6
Correct answer is option 'B'. Can you explain this answer?
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If the latus rectum of a hyperbola through one focus, subtends 60 °...
Taking only positive value of e as eccentricity cannot be negative.
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If the latus rectum of a hyperbola through one focus, subtends 60 °...
Degrees at the other focus, then the eccentricity of the hyperbola is 2.
Let's denote the distance between the foci of the hyperbola as 2a, and the length of the latus rectum as 2b.
The latus rectum of a hyperbola is defined as the line segment passing through one focus and perpendicular to the major axis, and it is given by the equation:
latus rectum = 2b = 2a^2 / c
where c is the distance between the center of the hyperbola and each focus.
Given that the latus rectum subtends 60 degrees at the other focus, we can say that the angle between the latus rectum and the major axis is 30 degrees.
Using the definition of the eccentricity of a hyperbola:
eccentricity = c / a
We can rewrite the equation for the latus rectum in terms of a and the eccentricity:
2b = 2a^2 / (eccentricity * a)
b = a / eccentricity
Since the latus rectum subtends 60 degrees at the other focus, we know that the distance between the center and each focus is a / sin(30 degrees). Therefore, c = a / sin(30 degrees).
Substituting this value into the equation for the latus rectum, we get:
b = a / (a / sin(30 degrees))
b = sin(30 degrees)
Using the definition of the eccentricity, we can write:
eccentricity = c / a = (a / sin(30 degrees)) / a = 1 / sin(30 degrees)
eccentricity = 2
Therefore, the eccentricity of the hyperbola is 2.
Let's denote the distance between the foci of the hyperbola as 2a, and the length of the latus rectum as 2b.
The latus rectum of a hyperbola is defined as the line segment passing through one focus and perpendicular to the major axis, and it is given by the equation:
latus rectum = 2b = 2a^2 / c
where c is the distance between the center of the hyperbola and each focus.
Given that the latus rectum subtends 60 degrees at the other focus, we can say that the angle between the latus rectum and the major axis is 30 degrees.
Using the definition of the eccentricity of a hyperbola:
eccentricity = c / a
We can rewrite the equation for the latus rectum in terms of a and the eccentricity:
2b = 2a^2 / (eccentricity * a)
b = a / eccentricity
Since the latus rectum subtends 60 degrees at the other focus, we know that the distance between the center and each focus is a / sin(30 degrees). Therefore, c = a / sin(30 degrees).
Substituting this value into the equation for the latus rectum, we get:
b = a / (a / sin(30 degrees))
b = sin(30 degrees)
Using the definition of the eccentricity, we can write:
eccentricity = c / a = (a / sin(30 degrees)) / a = 1 / sin(30 degrees)
eccentricity = 2
Therefore, the eccentricity of the hyperbola is 2.
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If the latus rectum of a hyperbola through one focus, subtends 60 ° angle at the other focus, then its eccentricity isa)√2b)√3c)√5d)√6Correct answer is option 'B'. Can you explain this answer?
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