Here, you will learn equation of asymptotes of hyperbola and how to find the asymptotes of the hyperbola and the director circle of hyperbola.
Let’s begin –
Equation of Asymptotes of hyperbola
If the length of the perpendicular let fall from the point on the hyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the Asymptote of the hyperbola.
How to find the asymptotes of the hyperbola :
Let y = mx + c is the asymptote of the hyperbola \(x^2\over a^2\) – \(y^2\over b^2\) = 1.
Solving these two we get the quadratic as \((b^2 – a^2m^2)\)\(x^2\) – 2\(a^2\)mcx – \(a^2(b^2 + c^2)\) = 0 …….(1)
In order that y = mx + c be an asymptote, both roots of equation (1) must approach infinity, the condition for which are :
coefficient of \(x^2\) = 0 & coefficient of x = 0.
\(\implies\) \((b^2 – a^2m^2)\) = 0 or m = \(\pm b\over a\) & \(a^2\)mc = 0 \(\implies\) c = 0.
\(\therefore\) equations of asymptote are \(x\over a\) + \(y\over b\) = 0 and \(x\over a\) – \(y\over b\) = 0
combined equation to the asymptotes \(x^2\over a^2\) – \(y^2\over b^2\) = 0
When b = a, the asymptotes of the rectangular hyperbola.
\(x^2 – y^2\) = \(a^2\) are y = \(\pm\)x which are at right angles.
Example : Find the asymptotes of the hyperbola \(2x^2 + 5xy + 2y^2 + 4x + 5y\) = 0
Solution : Let \(2x^2 + 5xy + 2y^2 + 4x + 5y + k\) = 0 be asymptotes. This will represent two straight line
so \(abc + 2fgh – af^2 – bg^2 – ch^2\) = 0 \(\implies\) 4k + 25 – \(25\over 2\) – 8 – \(25\over 4\)k = 0
\(\implies\) k = 2
\(\implies\) \(2x^2 + 5xy + 2y^2 + 4x + 5y + 2\) = 0 are asymptotes
\(\implies\) (2x+y+2) = 0 and (x+2y+1) = 0 are asymptotes