Question

Find the asymptotes of the following hyperbola and also the equations to the conjugate hyperbola: ${{y}^{2}}-xy-2{{x}^{2}}-5y+x-6=0$

Answer 1

Hint: Let the combined equation of the asymptotes be ${{y}^{2}}-xy-2{{x}^{2}}-5y+x+k=0$. Then factorize the second degree terms and find the combined equation of asymptotes. Equate the two equations to find the constants. Then use the formula: Equation of hyperbola + equation of conjugate hyperbola = 2 $\times$ (equation of asymptotes)

Complete step-by-step answer:

We are given the equation of the hyperbola: ${{y}^{2}}-xy-2{{x}^{2}}-5y+x-6=0$
We know that the equation of a hyperbola and the combined equation of the asymptotes are the same except for the constant term.
Therefore the combined equation of the asymptotes is of the form:
${{y}^{2}}-xy-2{{x}^{2}}-5y+x+k=0$, where k is a constant …(1)
Factorising the 2nd degree terms, we get the following:
${{y}^{2}}-xy-2{{x}^{2}}={{y}^{2}}-2xy+xy-2{{x}^{2}}=\left( y-2x \right)\left( y+x \right)$
Therefore the asymptotes have equations: $y-2x+l=0$ and $y+x+m=0$,
Where l and m are constants.
Their combined equation will be: $\left( y-2x+l \right)\left( y+x+m \right)=0$
${{y}^{2}}-xy-2{{x}^{2}}+\left( m+l \right)y+\left( -2m+l \right)x+ml=0$ …(2)
Equations (1) and (2) represent the same pair of lines.
Now, we will compare the coefficients of x, y, and the constant terms, we will get the following:
$m+l=-5$
$-2m+l=1$
$ml=k$
From the first two conditions: $m+l=-5$ and $-2m+l=1$, we will solve for m and l.
Subtract the two equations, we get:
3m = -6
m = -2
Putting this in $m+l=-5$, we get the following:
-2 + l = -5
l = -3
So, m = -2 and l = -3
And k = ml = 6
Therefore, the asymptotes are $y-2x+-3=0$ and $y+x+-2=0$, and their combined equation is ${{y}^{2}}-xy-2{{x}^{2}}-5y+x+6=0$
Now we know the fact that:
Equation of hyperbola + equation of conjugate hyperbola = 2 $\times$ (equation of asymptotes)
So, Equation of conjugate hyperbola = 2 $\times$ (equation of asymptotes) - equation of hyperbola
Equation of conjugate hyperbola = $2~\times \left( {{y}^{2}}-xy-2{{x}^{2}}-5y+x+6 \right)-\left( {{y}^{2}}-xy-2{{x}^{2}}-5y+x-6 \right)$
Hence, equation of conjugate hyperbola is ${{y}^{2}}-xy-2{{x}^{2}}-5y+x+18=0$

Note: In this question, it is important to know the various concepts of hyperbola like the equation of a hyperbola and the combined equation of the asymptotes are the same except for the constant term. You should also know the formula: Equation of hyperbola + equation of conjugate hyperbola = 2 $\times$ (equation of asymptotes)