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More # ${\text{Prove that the line }}lx + my + n = 0{\text{ touches the parabola }}{y^2}{\text{ = }}4a(x - b){\text{ if }}a{m^2} = b{l^2} + nl.$ Verified
${\text{straight line }}lx + my + n = 0{\text{ is tangent to}} \\ {\text{parabola }}{y^2} = 4a(x - b) \\ {\text{if line }}y = Mx + c{\text{ touches parabola }}{y^2} = 4a(x - b) \\ {\text{then}} \\ c = \dfrac{a}{M}{\text{ }}....................{\text{(1)}} \\ {\text{for the given line }}lx + my + n = 0 \\ l(x + b) + my + n = 0 \\ y = \dfrac{{ - l(x + b) - n}}{m} \\ y = \dfrac{{ - lx}}{m} + \dfrac{{ - lb - n}}{m}....................{\text{(2)}} \\ {\text{compare equation (2) with the equation }}y = Mx + c \\ M = \dfrac{{ - l}}{m},c = \dfrac{{ - lb - n}}{m} \\ {\text{put these value in the equation (1) the equation become }} \\ \dfrac{{ - lb - n}}{m} = \dfrac{a}{{\dfrac{{ - l}}{m}}} \\ \dfrac{{ - lb - n}}{m} = \dfrac{{am}}{{ - l}} \\ l{b^2} + nl = a{m^2}.{\text{ Hence Proved}}{\text{.}} \\ {\text{Note: If }}l{b^2} + nl = a{m^2}{\text{ then the line }}lx + my + n = 0{\text{ will touches the parabola }} \\ {y^2} = 4a(x - b). \\$