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10-2. Parabola, Ellipse, Hyperbola
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રેખા $\,y\,\, = \,\,ax\,\, + \;\,b$ એ અતિવલય $\,\,\frac{{{x^2}}}{{{a^2}}}\,\, - \,\,\frac{{{y^2}}}{{{b^2}}}\,\, = \,\,1$ નો સ્પર્શક હોવાથી શરત હેઠળ ગતિ કરતા બિંદુ $P\,\,\left( {a,\,\,b} \right)\,\,$ નો બિંદુપથ
A
અતિવલય
B
પરવલય
C
વર્તૂળ
D
ઉપવલય
Solution
Given, line $y=\alpha x+\beta$ is a tangent to the given hyperbola, if $\beta^2=a^2 \alpha^2-b^2$.
Hence, locus of $(\alpha, \beta)$ is $y^2=a^2 x^2-b^2$
$\Rightarrow \frac{y^2}{b^2}=\frac{a^2 x^2}{b^2}-1$
$\Rightarrow \frac{x^2}{b^2 / a^2}-\frac{y^2}{b^2}=1 \text { which is hyperbola }$
Standard 11
Mathematics
