Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola \(5y^ 2 - 9x^ 2 = 36\)
Solution and Explanation
The given equation is \(5y^ 2 - 9x ^2 = 36\)
\(⇒ \frac{5y^2}{36} –\frac{ 9x^2}{36} = \frac{36}{36}\)
\(\frac{y^2}{(\frac{36}{5})} – \frac{x^2}{4} = 1.......(1)\)
On comparing equation (1) with the standard equation of hyperbola i.e., \(\frac{y^2}{a^2} –\frac{ x^2}{b^2} = 1\), we obtain \(a= \frac{6}{\sqrt5}\) and b = 2
We know that \(a ^2 + b^ 2 = c^ 2 .\)
\(∴ c^2 = \frac{36}{5} + 4\)
\(c^2 = \frac{56}{5}\)
\(c = \sqrt{(\frac{56}{5})}\)
\(= \frac{2\sqrt{14}}{\sqrt{5}}\)
Therefore,
The coordinates of the foci are (0, \(\frac{2\sqrt{14}}{\sqrt{5}}\)) and (0, – \(\frac{2\sqrt{14}}{\sqrt{5}}\)).
The coordinates of the vertices are (0, \(\frac{6}{\sqrt5}\)) and (0, -\(\frac{6}{\sqrt5}\)).
Eccentricity, \(e = \frac{c}{a} =\frac{ (\frac{2\sqrt{14}}{\sqrt{5}}) }{ (\frac{6}{\sqrt5}) }= \frac{\sqrt{14}}{3}\)
Length of latus rectum = \(\frac{2b^2}{a} =\frac{(2 \times 22)}{\frac{6}{\sqrt{5}}} = \frac{(2\times4)}{\frac{6}{\sqrt{5 }}}= \frac{4\sqrt5}{3}\)
Top Questions on Hyperbola
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- Consider a hyperbola \( H \) having its centre at the origin and foci on the \( x \)-axis. Let \( C_1 \) be the circle touching the hyperbola \( H \) and having its centre at the origin. Let \( C_2 \) be the circle touching the hyperbola \( H \) at its vertex and having its centre at one of its foci. If the areas (in square units) of \( C_1 \) and \( C_2 \) are \( 36\pi \) and \( 4\pi \), respectively, then the length (in units) of the latus rectum of \( H \) is:
- Let H: $\frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha, 6)$, $\alpha>0$ lies on H. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$, then $\alpha^2 + \beta$ is equal to:
- If the hyperbola \(x² - y² cosec²θ = 5\) and ellipse \(x²cosec²θ + y² = 5\) has eccentricity \(e_H\) and \(e_E\) respectively and \(e_H = \sqrt 7e_E\), then \(θ\) is equal to
- The foci of a hyperbola are (±2,0) and its eccentricity is \(\frac{3}{2}\) . A tangent, perpendicular to the line 2x + 3y = 6, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x- and y-axes are a and b respectively, then |6a| + |5b| is equal to_____.
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Concepts Used:
Hyperbola
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.
