Geo3.23

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The equation ax^{2}+2hxy+by^{2}+2gx+2fy+c=0\, represents a pair of parallel lines. Prove that the equation of the line midway between the two parallel lines is hx+by+f=0\,

Let the lines be lx+my+n_{1}=0,lx+my+n_{2}=0\, let these equations be 1 and 2

(lx+my+n_{1})(lx+my+n_{2})\equiv ax^{2}+2hxy+by^{2}+2gx+2fy+c\,

l^{2}=a,m^{2}=b,lm=h,l(n_{1}+n_{2})=2g,m(n_{1}+n_{2})=2f\,

The equation to the line midway between the two lines is

{\frac  {1}{2}}[(lx+my+n_{1})+(lx+my+n_{2})]=0\,

lx+my+{\frac  {n_{1}+n_{2}}{a}}=0\,

Multiplying by m,we have lmx+m^{2}y+{\frac  {m}{2}}(n_{1}+n_{2})=0\,

Therefore the condition is hx+by+f=0\,


Main Page:Geometry:Straight Lines-II