JEE Advance - Mathematics (1988 - No. 22)
Lines$${L_1} = ax + by + c = 0$$ and $${L_2} = lx + my + n = 0$$ intersect at the point $$P$$ and make an angle $$\theta $$ with each other. Find the equation of a line $$L$$ different from $${L_2}$$ which passes through $$P$$ and makes the same angle $$\theta $$ with $${L_1}$$.
$$\left( {{a^2} - {b^2}} \right)\left( {\ell x + my + n} \right) - \left( {a\ell + bm} \right)left( {ax + by + c} \right) = 0$$
$$\left( {{a^2} + {b^2}} \right)\left( {\ell x - my + n} \right) - 2\left( {a\ell + bm} \right)left( {ax + by + c} \right) = 0$$
$$\left( {{a^2} + {b^2}} \right)\left( {\ell x + my + n} \right) + 2\left( {a\ell + bm} \right)left( {ax + by + c} \right) = 0$$
$$\left( {{a^2} + {b^2}} \right)\left( {\ell x + my + n} \right) - 2\left( {a\ell + bm} \right)left( {ax + by + c} \right) = 0$$
$$\left( {{a^2} - {b^2}} \right)\left( {\ell x + my + n} \right) + \left( {a\ell + bm} \right)left( {ax + by + c} \right) = 0$$
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