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Find the equations to the tangents to the ellipse 4x^{2}+3y^{2}=5\, which are parallel to 3x-y+7=0\,

Given ellipse is 4x^{2}+3y^{2}=5\,

Equaton of the tangent is y=mx\pm {\sqrt  {a^{2}m^{2}+b^{2}}}\,

y=mx+{\sqrt  {{\frac  {5}{4}}m^{2}+{\frac  {5}{3}}}}\,

Since the tangent is parallel to the line 3x-y+7=0\,,the slope of the required line equal to the slope of the given line.

Therefore m=3.

Hence the equation of the required tangent is y=3x\pm {\sqrt  {{\frac  {45}{4}}+{\frac  {5}{3}}}}\,

y=3x\pm {\sqrt  {{\frac  {155}{12}}}}\,

Main Page:Geometry:The Ellipse