The condition that the line ax + by + c = 0 i | vedclass

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(C) The equation of the line is $ax + by + c = 0$,which can be rewritten as $y = -\frac{a}{b}x - \frac{c}{b}$.For a line $y = mx + k$ to be tangent to

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$b^2 = ac$

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(C) The equation of the line is $ax + by + c = 0$,which can be rewritten as $y = -\frac{a}{b}x - \frac{c}{b}$.For a line $y = mx + k$ to be tangent to the parabola $y^2 = 4Ax$ (where $A=a$),the condition is $k = \frac{A}{m}$.Here,$m = -\frac{a}{b}$ and $k = -\frac{c}{b}$.Substituting these into the condition: $-\frac{c}{b} = \frac{a}{-\frac{a}{b}}$.$-\frac{c}{b} = -\frac{ab}{a} = -b$.Therefore,$c = b^2$ is incorrect based on standard derivation; the correct condition is $c = ab^2/a = b^2$ is not standard. Let us re-evaluate: $k = A/m$ $\Rightarrow -c/b = a/(-a/b)$ $\Rightarrow -c/b = -b$ $\Rightarrow c = b^2$. Wait,the standard form $y^2=4ax$ with line $y=mx+c'$ gives $c'=a/m$. Here $m=-a/b$ and $c'=-c/b$. So $-c/b = a/(-a/b) = -b$. Thus $c = b^2$. The correct option is $c = b^2$.

The condition that the line $ax + by + c = 0$ is tangent to the parabola $y^2 = 4ax$ is:

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