The locus of the point of intersection of the | vedclass

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(C) Given lines are:$L_1: \sqrt{3}x - y = 4\sqrt{3}k$  $(i)$$L_2: \sqrt{3}kx + ky = 4\sqrt{3}$  (ii)From $(i)$,$k = \frac{\sqrt{3}x - y}{4\sqrt{3}}$.S

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Hyperbola

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(C) Given lines are:$L_1: \sqrt{3}x - y = 4\sqrt{3}k$  $(i)$$L_2: \sqrt{3}kx + ky = 4\sqrt{3}$  (ii)From $(i)$,$k = \frac{\sqrt{3}x - y}{4\sqrt{3}}$.Substitute $k$ into (ii):$\sqrt{3}x(\frac{\sqrt{3}x - y}{4\sqrt{3}}) + y(\frac{\sqrt{3}x - y}{4\sqrt{3}}) = 4\sqrt{3}$Multiply by $4\sqrt{3}$:$\sqrt{3}x(\sqrt{3}x - y) + y(\sqrt{3}x - y) = 48$$3x^2 - \sqrt{3}xy + \sqrt{3}xy - y^2 = 48$$3x^2 - y^2 = 48$Dividing by $48$:$\frac{x^2}{16} - \frac{y^2}{48} = 1$This is the equation of a hyperbola.

The locus of the point of intersection of the lines $\sqrt{3}x - y - 4\sqrt{3}k = 0$ and $\sqrt{3}kx + ky - 4\sqrt{3} = 0$ for different values of $k$ is

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