The line lx + my + n = 0 will be a tangent to | vedclass

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(A) The condition for the line $y = Mx + C$ to be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $C^2 = a^2M^2 - b^2$.Given the

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$a^2l^2 - b^2m^2 = n^2$

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(A) The condition for the line $y = Mx + C$ to be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $C^2 = a^2M^2 - b^2$.Given the line $lx + my + n = 0$,we can rewrite it as $y = -\frac{l}{m}x - \frac{n}{m}$.Comparing this with $y = Mx + C$,we get $M = -\frac{l}{m}$ and $C = -\frac{n}{m}$.Substituting these values into the condition $C^2 = a^2M^2 - b^2$:$(-\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 get $n^2 = a^2l^2 - b^2m^2$,which can be written as $a^2l^2 - b^2m^2 = n^2$.

The line $lx + my + n = 0$ will be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,if

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