Foci (0, \(±\sqrt{10}\)) and passing through (2, 3)
Here, the foci are on the y-axis.
Therefore, the equation of the hyperbola is of the form \(\frac{y^2}{a^2} –\frac{ x^2}{b^2} = 1\)
Since the foci are\( (±\sqrt{10}, 0), c = \sqrt{10}\)
We know that \(a^2 + b^2 = c^2\)
\(∴ b^2 = 10 – a^2 ………….. (1)\)
Since the hyperbola passes through point (2, 3),
\(\frac{9}{a^2} – \frac{4}{b^2} = 1 … (2)\)
From equations (1) and (2), we obtain
\(\frac{9}{a^2} – \frac{4}{(10-a^2)} = 1\)
\(⇒ 9(10 – a^2) – 4a^2 = a^2(10 –a^2)\)
\(⇒ 90 – 9a^2 – 4a^2 = 10a^2 – a^4\)
\(⇒ a^4 – 23a^2 + 90 = 0\)
\(⇒ a^4 – 18a^2 – 5a^2 + 90 = 0\)
\(⇒ a^2(a^2 -18) -5(a^2 -18) = 0\)
\(⇒ (a^2 – 18) (a^2 -5) = 0\)
\(⇒ a^2 = 18\) or \(5\)
In hyperbola, \(c > a, i.e., c^2 > a^2\)
\(∴ a^2 = 5\)
\(⇒ b^2 = 10 – a^2= 10 – 5= 5\)
Thus, the equation of the hyperbola is \(\frac{y^2}{5} – \frac{x^2}{5} = 1\)
Hyperbola is the locus of all the points in a plane such that the difference in their distances from two fixed points in the plane is constant.
Hyperbola is made up of two similar curves that resemble a parabola. Hyperbola has two fixed points which can be shown in the picture, are known as foci or focus. When we join the foci or focus using a line segment then its midpoint gives us centre. Hence, this line segment is known as the transverse axis.