Given, ax2+2hxy+by2+2gx+2fy+c=0 .......(i) This represents pair of straight lines when abc+2fgh−af2−bg2−ch2=0......(ii) When the pair of lines intersect X-axis, y=0. Put y=0 in Eq. (i), ax2+2gx+c=0 This is a quadratic equation in x. Since lines meet at X-axis, the roots must be equal. So, b2−4ac=0 ⇒4g2−4ac=0⇒g2=ac Put g2=ac in Eq. (ii), we get abc+2fgh−af2−bac−ch2=0 ⇒2fgh=af2+ch2.......(iii) Eq. (iii) is possible when f=h and a+c=2g