The line y = mx + c touches the curve frac{x^ | vedclass

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(B) 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 derived by substituting $y = m

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$c^2 = a^2m^2 - b^2$

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(B) 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 derived by substituting $y = mx + c$ into the equation of the hyperbola.Substituting $y$ gives $\frac{x^2}{a^2} - \frac{(mx + c)^2}{b^2} = 1$.Multiplying by $a^2b^2$,we get $b^2x^2 - a^2(m^2x^2 + 2mcx + c^2) = a^2b^2$.Rearranging as a quadratic in $x$: $(b^2 - a^2m^2)x^2 - (2a^2mc)x - (a^2c^2 + a^2b^2) = 0$.Since the line is a tangent,the discriminant $D = 0$.$D = (2a^2mc)^2 - 4(b^2 - a^2m^2)(-a^2c^2 - a^2b^2) = 0$.Simplifying this leads to the condition $c^2 = a^2m^2 - b^2$.

The line $y = mx + c$ touches the curve $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,if

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