Here you will learn formula to find the equation of directrix of hyperbola with examples.
Let’s begin –
Directrix of Hyperbola Equation
(i) For the hyperbola \(x^2\over a^2\) – \(y^2\over b^2\) = 1
The equation of directrix is x = \(a\over e\) and x = \(-a\over e\)
(ii) For the hyperbola -\(x^2\over a^2\) + \(y^2\over b^2\) = 1
The equation of directrix is y = \(b\over e\) and y = \(-b\over e\)
Also Read : Equation of the Hyperbola | Graph of a Hyperbola
Example : For the given ellipses, find the equation of directrix.
(i) \(16x^2 – 9y^2\) = 144
(ii) \(9x^2 – 16y^2 – 18x + 32y – 151\) = 0
Solution :
(i) We have,
\(16x^2 – 9y^2\) = 144 \(\implies\) \(x^2\over 9\) – \(y^2\over 16\) = 1,
This is of the form \(x^2\over a^2\) – \(y^2\over b^2\) = 1
where \(a^2\) = 9 and \(b^2\) = 16 i.e. a = 3 and b = 4
Eccentricity (e) = \(\sqrt{1 + {b^2\over a^2}}\)
\(\implies\) e = \(\sqrt{1 + 16/9}\) = \(5\over 3\)
Therefore, the equation of directrix is x = \(\pm {a\over e}\)
\(\implies\) x = \(\pm {9\over 5}\)
(ii) We have,
\(9x^2 – 16y^2 – 18x + 32y – 151\) = 0
\(\implies\) \(9(x^2 – 2x)\) – \(16(y^2 – 2y)\) = 151
\(\implies\) \(9(x^2 – 2x + 1)\) – \(16(y^2 – 2y + 1)\) = 144
\(\implies\) \(9(x – 1)^2\) – \(16(y – 1)^2\) = 144
\(\implies\) \((x – 1)^2\over 16\) – \((y – 1)^2\over 9\) = 1
Here, a = 4 and b = 3
Eccentricity (e) = \(\sqrt{1 + {b^2\over a^2}}\)
\(\implies\) e = \(\sqrt{1 + 9/16}\) = \(5\over 4\)
Here, the center is (h, k) i.e. (1, 1)
Therefore, the equation of directrix is x = \(\pm {a\over e}\) + h
\(\implies\) x = \(\pm {16\over 5}\) + 1
\(\implies\) x= \(21\over 5\) and x = \(-11\over 5\)
Note : For the hyperbola \((x – h)^2\over a^2\) – \((y – k)^2\over b^2\) = 1 with center (h. k),
(i) For normal hyperbola,
The equation of directrix is x = \(\pm {a\over e}\) + h
(ii) For conjugate hyperbola,
The equation of directrix is y = \(\pm {b\over e}\) + k
