Find the equation of the lines joining the origin to the points of intersection of the straight line y=3x+2 with the curve x2+2xy+3y2+4x+8y−11=0
7x2−2xy−y2=0
We will use method of homogenization to solve this. The combination of pairs straight lines is given by
Ax2+2hxy+by2+2gx(lx+my−n)+2fy(lx+my−n)+c(lx+my−n)2=0
the term equivalent to lx+my−n is y−3x2. We can take (3x−y−2) also.
⇒x2+2xy+3y2+4x(y−3x2)+8x(y−3x2)-11x(y−3x2)2
⇒x2+2xy+3y2+2xy−6x2+4y2−12xy−114(y2+9x2−6xy)=0
⇒−5x2+7y2−8xy−11y2−99x2+66xy4=0
⇒−20x2+28y2−32xy−11y2−99x2+66xy=0
⇒−119x2+17y2+34xy=0
⇒7x2−2xy−y2=0