Geo3.20

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If 2x^{2}+3xy+py^{2}-5x+10y+q=0\, represents a pair of perpendicular lines,find p,q.Find their point of intersection.

From the given equation a=2,b=p,c=q,h={\frac  {3}{2}},g={\frac  {-5}{2}},f=5\,

abc+2fgh-af^{2}-bg^{2}-ch^{2}=0,a+b=0,2+p=0,p=-2\,

-16q-150-200+50-9q=0,25q=-300,q=-12\,

The values of p and q are -2 and -12 respectively.

Also the point of intersection is \left({\frac  {{\frac  {15}{2}}-5}{-4-{\frac  {9}{4}}}},{\frac  {({\frac  {-15}{4}})-10}{-4-{\frac  {9}{4}}}}\right)\,

({\frac  {-2}{5}},{\frac  {11}{5}})\,


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