To trace the curve, find asymptotes, and identify the region of absence for the equation xy^2 = a(a - x), we'll follow these steps:
1. Find the asymptotes:
The equation xy^2 = a(a - x) can be rearranged to isolate y^2:
y^2 = (a(a - x))/x
Now, find the asymptotes by considering the limits as x approaches infinity and as x approaches zero:
As x approaches infinity:
y^2 ≈ a(a - x)/x
Since x is much larger than a in this limit, y^2 ≈ -a, which means there are horizontal asymptotes at y = ±√(-a).
As x approaches zero:
y^2 ≈ a(a - x)/x
In this limit, x dominates, and y^2 ≈ -ax/x = -a. So, there's a horizontal asymptote at y = √(-a) and another one at y = -√(-a).
2. Region of Absence:
To determine the region of absence, consider the original equation xy^2 = a(a - x). The term y^2 must be non-negative, which means that for real solutions, a(a - x) must also be non-negative. When is this not satisfied?
a(a - x) is non-negative when:
- Both a and (a - x) are non-negative (i.e., a ≥ 0 and a - x ≥ 0), which gives us 0 ≤ x ≤ a.
- Or both a and (a - x) are non-positive (i.e., a ≤ 0 and a - x ≤ 0), which gives us x ≥ a.
Now, you have the asymptotes at y = ±√(-a) and the region of absence as x ≤ 0 and x ≥ a for the curve described by the equation xy^2 = a(a - x).
Answers & Comments
Verified answer
Answer:
the region of absence is x ≤ 0 and x ≥ a.
Step-by-step explanation:
To trace the curve, find asymptotes, and identify the region of absence for the equation xy^2 = a(a - x), we'll follow these steps:
1. Find the asymptotes:
The equation xy^2 = a(a - x) can be rearranged to isolate y^2:
y^2 = (a(a - x))/x
Now, find the asymptotes by considering the limits as x approaches infinity and as x approaches zero:
As x approaches infinity:
y^2 ≈ a(a - x)/x
Since x is much larger than a in this limit, y^2 ≈ -a, which means there are horizontal asymptotes at y = ±√(-a).
As x approaches zero:
y^2 ≈ a(a - x)/x
In this limit, x dominates, and y^2 ≈ -ax/x = -a. So, there's a horizontal asymptote at y = √(-a) and another one at y = -√(-a).
2. Region of Absence:
To determine the region of absence, consider the original equation xy^2 = a(a - x). The term y^2 must be non-negative, which means that for real solutions, a(a - x) must also be non-negative. When is this not satisfied?
a(a - x) is non-negative when:
- Both a and (a - x) are non-negative (i.e., a ≥ 0 and a - x ≥ 0), which gives us 0 ≤ x ≤ a.
- Or both a and (a - x) are non-positive (i.e., a ≤ 0 and a - x ≤ 0), which gives us x ≥ a.
Now, you have the asymptotes at y = ±√(-a) and the region of absence as x ≤ 0 and x ≥ a for the curve described by the equation xy^2 = a(a - x).
Answer:
Trace the curve and find asymtotes and Region of absence.
Trace the curve and find asymtotes and Region of absence.xy^2=a(a-x).
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