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  • The asymptotes of the hyperbola hx + ky = xy are Option: 1 <img alt="x-k=0, y-h=0" src="https://learn.careers360.com/latex-image/?x-k%3D0%2C%20y

The asymptotes of the hyperbola hx + ky = xy are 

Option: 1

x-k=0, y-h=0


Option: 2

x+h=0, y+k=0


Option: 3

x-h=0, y-k=0


Option: 4

x+k=0, y+h=0


Answers (1)

best_answer

 

 

Asymptotes of Hyperbola -

Asymptotes of Hyperbola:

Asymptote of a curve is a straight line such that the distance between the curve and the line approaches to zero when one or both of x- and y- coordinate approach infinity.

For example,Asymptote of the curve y = 1/x is straight line y = 0 and x  = 0.

 

An asymptote of any hyperbola is a straight line that touches it at infinity.


 

\mathbf{\\\text{The equation of the asymptotes of the hyperbola }\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \\\mathrm{are\;\;y=\pm \frac{b}{a} x\;\;or\;\;\frac{x}{a}\pm\frac{y}{b}=0} }


 

To find the asymptotes of the hyperbola,

 

\\ {\text {Let the straight line } y=m x+c \text { is asymptotes to the hyperbola }} \\ {\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1} \\\\ {\text { put the value of } \mathit{y} \text { in the Eq. of hyperbola }} \\\\ {\frac{x^{2}}{a^{2}}-\frac{(m x+c)^{2}}{b^{2}}=1} \\\\ {\left.\text { (a }^{2} m^{2}-b^{2}\right) x^{2}+2 a^{2} m c x+a^{2}\left(c^{2}+b^{2}\right)=0} \\\\ {\text { since, asymptotes touch the hyperbola at infinity, so both roots of the }} \\ {\text { quadratic equation must be infinite }}\\\text{ and condition for which is coefficient of }x^2\text{ and }x\text{ must be zero.}

\\ {\therefore \;\;\;\;\;\;\;\;\;\;\mathrm{a}^{2} \mathrm{m}^{2}-\mathrm{b}^{2}=0} \\ {\text { and }\;\;\;\;\;\;\;2 \mathrm{a}^{2} \mathrm{mc}=0} \\\\ {\mathrm{m}=\pm \frac{\mathrm{b}}{\mathrm{a}} \text { and } \mathrm{c}=0} \\ {\text { put the value of } \mathrm{m} \text { in } \mathrm{y}=\mathrm{mx}+\mathrm{c}} \\\\ {\mathrm{y}=\pm \frac{\mathrm{b}}{\mathrm{a}} \mathrm{x} \;\;\;\;\text { or } \;\;\;\;\;\frac{\mathrm{x}}{\mathrm{a}} \pm \frac{\mathrm{y}}{\mathrm{b}}=0}

 

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 Given equation of the hyperbola is hx + ky = xy

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;} hx + ky = xy \\\Rightarrow \mathrm{\;\;\;\;\;\;\;\;} xy-ky=hx \\\Rightarrow \mathrm{\;\;\;\;\;\;\;\;} y(x-k)=hx \\\Rightarrow \mathrm{\;\;\;\;\;\;\;\;} \frac{y}{x}(x-k)=h

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;} \lim _{x\rightarrow \infty}\frac{y}{x}(x-k)=h\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;} \lim _{x\rightarrow \infty}y(1-\frac{k}{x})=h\\\Rightarrow y=h

Now, 

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;} \lim _{y\rightarrow \infty}\frac{y}{x}(x-k)=h\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;} \lim _{y\rightarrow \infty}\frac{hx}{y}=x-k\\\Rightarrow x=k

Equation of asymptotes are x = k and y = h

Posted by

Divya Prakash Singh

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