Geo3.22

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Find k if the equation 4x^{2}+2lxy+y^{2}+6x+ky=10\, represent a pair of parallel lines.Also find the distance between them.

From the given equation a=4,h=l,b=1,g=3,f={\frac  {k}{2}},c=-10\,

h^{2}=ab,l^{2}=4(1),l=\pm 2\,

bg^{2}=af^{2},(1)(9)=4[{\frac  {k^{2}}{4}}],k^{2}=9,k=\pm 3\,

Distance between the parallel lines is 2{\sqrt  {{\frac  {g^{2}-ac}{a(a+b)}}}}\,

The distance is 2{\sqrt  {{\frac  {3^{2}-4(-10)}{4(4+1)}}}}={\frac  {7}{{\sqrt  {5}}}}\,


Main Page:Geometry:Straight Lines-II