When is the line ℓx + my + n = 0 a tangent to | vedclass

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(A) The line $y = mx + c$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ if and only if $c^2 = a^2m^2 - b^2$.For the given line

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$a^2ℓ^2 - b^2m^2 = n^2$

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(A) The line $y = mx + c$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ if and only if $c^2 = a^2m^2 - b^2$.For the given line $ℓx + my + n = 0$,we can rewrite it as $y = -(\frac{ℓ}{m})x - (\frac{n}{m})$.Here,the slope is $M = -\frac{ℓ}{m}$ and the intercept is $C = -\frac{n}{m}$.Substituting these into the condition $C^2 = a^2M^2 - b^2$,we get:$(-\frac{n}{m})^2 = a^2(-\frac{ℓ}{m})^2 - b^2$$\frac{n^2}{m^2} = \frac{a^2ℓ^2}{m^2} - b^2$Multiplying both sides by $m^2$,we obtain $n^2 = a^2ℓ^2 - b^2m^2$,which can be rearranged as $a^2ℓ^2 - b^2m^2 = n^2$.

When is the line $ℓx + my + n = 0$ a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$?

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