If a>0 and b>0, then the eccentricity of the hyperbola ((x-h)^2)/(a^2)-((y-k)^2)/(b^2)=1 or

step 1 Hello everyone today we are going to solve problem number 58 here we have to find the eccentrici

If a>0 and b>0, then the eccentricity of the hyperbola...

If $a>0$ and $b>0,$ then the eccentricity of the hyperbola $$\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1$$ is the number $\frac{\sqrt{a^{2}+b^{2}}}{a} .$ Find the eccentricity of the hyperbola whose equation is given. $$\frac{y^{2}}{18}-\frac{x^{2}}{25}=1$$

If $a>0$ and $b>0,$ then the eccentricity of the hyperbola
$$\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1 \quad \text { or } \quad \frac{(y-k)^{2}}{a^{2}}-\frac{(x-h)^{2}}{b^{2}}=1$$
is the number $\frac{\sqrt{a^{2}+b^{2}}}{a} .$
Find the eccentricity of the hyperbola whose equation is given.
$$\frac{y^{2}}{18}-\frac{x^{2}}{25}=1$$

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Key Concepts

Hyperbola Eccentricity

The eccentricity of a hyperbola is a measure of how 'stretched' it is, defined as e = sqrt(a² + b²)/a when a is taken as the distance along the transverse axis. Since eccentricity is always greater than 1 for a hyperbola, it quantifies the degree of deviation from a perfect circle; the larger the eccentricity, the more open or elongated the hyperbola is.

Hyperbola

A hyperbola is a type of conic section that is defined as the set of all points in a plane such that the absolute difference of the distances from any point on the hyperbola to two fixed points (foci) is constant. Hyperbolas have two disconnected branches and are characterized by their centers, vertices, foci, and asymptotes, which provide a complete description of their geometric properties.

Standard Form of a Hyperbola

The standard form of a hyperbola's equation is used to simplify the analysis and graphing of hyperbolas. It typically takes one of two forms depending on the orientation: (x-h)²/a² - (y-k)²/b² = 1 for a horizontal transverse axis or (y-k)²/a² - (x-h)²/b² = 1 for a vertical transverse axis, where (h, k) is the center, a is the distance from the center to a vertex along the transverse axis, and b is related to the conjugate axis.

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