Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$
Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$
The given equation is $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ or $\frac{x^{2}}{4^{2}}-\frac{y^{2}}{3^{2}}=1$
On comparing this equation with the standard equation of hyperbola i.e., $\frac{ x ^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1,$ we obtain $a=4$ and $b=3$.
We known that $a^{2}=b^{2}+c^{2}$
$\therefore c^{2}=4^{2}+3^{2}=25$
$\Rightarrow c=5$
Therefore,
The coordinates of the foci are $(±5,\,0)$
The coordinates of the vertices are $(±4,\,0)$
Eccentricity, $e=\frac{c}{a}=\frac{5}{4}$
Length of latus rectum $=\frac{2 b^{2}}{a}=\frac{2 \times 9}{4}=\frac{9}{2}$
Similar Questions
If the tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercept $a_1, a_2$ on $x-$ axis and $b_1, b_2$ on $y-$ axis, then $(a_1a_2 + b_1b_2)$ is
If the tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercept $a_1, a_2$ on $x-$ axis and $b_1, b_2$ on $y-$ axis, then $(a_1a_2 + b_1b_2)$ is
The line $2 \mathrm{x}+\mathrm{y}=1$ is tangent to the hyperbola $\frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$. If this line passes through the point of intersection of the nearest directrix and the $\mathrm{x}$-axis, then the eccentricity of the hyperbola is
The line $2 \mathrm{x}+\mathrm{y}=1$ is tangent to the hyperbola $\frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$. If this line passes through the point of intersection of the nearest directrix and the $\mathrm{x}$-axis, then the eccentricity of the hyperbola is
- [IIT 2010]
For hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1$ which of the following remain constant if $\alpha$ varies
For hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1$ which of the following remain constant if $\alpha$ varies
Let $\lambda x-2 y=\mu$ be a tangent to the hyperbola $a^{2} x^{2}-y^{2}=b^{2}$. Then $\left(\frac{\lambda}{a}\right)^{2}-\left(\frac{\mu}{b}\right)^{2}$ is equal to
Let $\lambda x-2 y=\mu$ be a tangent to the hyperbola $a^{2} x^{2}-y^{2}=b^{2}$. Then $\left(\frac{\lambda}{a}\right)^{2}-\left(\frac{\mu}{b}\right)^{2}$ is equal to
- [JEE MAIN 2022]
The locus of middle points of the chords of the circle $x^2 + y^2 = a^2$ which touch the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is
The locus of middle points of the chords of the circle $x^2 + y^2 = a^2$ which touch the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is