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

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Question

If $y = Mx + C$ is tangent to hyperbola $\frac{ x ^2}{ a ^2}-\frac{ y ^2}{ b ^2}=1$, then $C ^2= a ^2 M ^2- b ^2$.

$lx + my + n =0 	o y =-\frac{1}

Vedclass · Accepted Answer

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

Vedclass · Answer

If $y = Mx + C$ is tangent to hyperbola $\frac{ x ^2}{ a ^2}-\frac{ y ^2}{ b ^2}=1$, then $C ^2= a ^2 M ^2- b ^2$.

$lx + my + n =0 	o y =-\frac{1}{ m } x -\frac{ n }{ m }$

Comparing this equation with $y = Mx + C$, we get

$M=-\frac{1}{ m }, C =-\frac{ n }{ m }$

Now, $C ^2= a ^2 M ^2- b ^2$

$\frac{ n ^2}{ m ^2}= a ^2 \frac{1^2}{ m ^2}- b ^2$

$\Rightarrow n ^2= a ^2 1^2- b ^2 m ^2$

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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