A tangent to a hyperbola frac{x^2}{a^2} - fra | vedclass

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(B) The equation of the tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ at the point $(a \sec 	heta, b 	an 	heta)$ is given by $\f

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$x^2 - y^2 = 1$

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(B) The equation of the tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ at the point $(a \sec 	heta, b 	an 	heta)$ is given by $\frac{x \sec 	heta}{a} - \frac{y 	an 	heta}{b} = 1$.This can be rewritten as $\frac{x}{a \cos 	heta} - \frac{y}{b \cot 	heta} = 1$.The intercepts on the coordinate axes are $x_0 = a \cos 	heta$ and $y_0 = -b \cot 	heta$.Given that the intercepts have a length of unity,we have $|a \cos 	heta| = 1$ and $|-b \cot 	heta| = 1$.Thus,$a = \frac{1}{|\cos 	heta|} = |\sec 	heta|$ and $b = \frac{1}{|\cot 	heta|} = |	an 	heta|$.Squaring and subtracting,we get $a^2 - b^2 = \sec^2 	heta - 	an^2 	heta = 1$.Therefore,the point $(a, b)$ lies on the rectangular hyperbola $x^2 - y^2 = 1$.

$A$ tangent to a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ intercepts a length of unity from each of the coordinate axes. Then the point $(a, b)$ lies on the rectangular hyperbola:

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