AP EAPCET 04-Jul-2022 Shift 1 Paper

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