The hyperbola with equation $xy = 8$ is the hyperbola $\frac{x^2 - y^2}{k} = 1$ referred to different axes - HSC - SSCE Mathematics Extension 2 - Question 7 - 2016 - Paper 1
Question 7
The hyperbola with equation $xy = 8$ is the hyperbola $\frac{x^2 - y^2}{k} = 1$ referred to different axes.
What is the value of $k$?
(A) 2
(B) 4
(C) 8
(D) 16
Worked Solution & Example Answer:The hyperbola with equation $xy = 8$ is the hyperbola $\frac{x^2 - y^2}{k} = 1$ referred to different axes - HSC - SSCE Mathematics Extension 2 - Question 7 - 2016 - Paper 1
Step 1
Determine the given hyperbola
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Answer
The equation given is xy=8. We can rewrite this in the standard form of a hyperbola. Notice that xy=c corresponds to the form kx2−y2=1. Looking at xy=8, we can factor this expression.
Step 2
Convert to standard form
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Answer
To convert xy=8 into the given hyperbola format, we can express it as:
8x2−8y2=1
This reflects the standard form definition of a hyperbola.
Step 3
Identify the value of k
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Answer
From the comparison of standard forms, we can identify that:
k=8
However, since we need kx2−y2=1 in the given format, we actually have:
8x2−y2=1
Therefore, the value of k must be 16.
