If the pair of lines ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 intersect on the y-axis then

Question

If the pair of lines ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 intersect on the y-axis then (A) 2fgh = bg2 + ch2 (B) bg2 ≠ ch2 (C) abc = 2fgh (D) none of these

Tardigrade Admin · Accepted Answer

Put x = 0 in the given equation ⇒ by2 + 2 fy + c = 0. For unique point of intersection f2- bc = 0 ⇒ af2 - abc = 0. Since abc + 2fgh - af2 - bg2 - ch2 = 0 ⇒ 2fgh - bg2 - ch2 = 0

Q. If the pair of lines $a x^{2} + 2 h x y + b y^{2} + 2 gx + 2 f y + c = 0$ intersect on the y-axis then

$2 f g h = b g^{2} + c h^{2}$

$b g^{2} \neq = c h^{2}$

$ab c = 2 f g h$

none of these

Put x = 0 in the given equation
$\Rightarrow b y^{2} + 2 f y + c = 0$ .
For unique point of intersection $f^{2} - b c = 0$
$\Rightarrow a f^{2} - ab c = 0$ .
Since $ab c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$
$\Rightarrow 2 f g h - b g^{2} - c h^{2} = 0$

Q. If the pair of lines ax2+2hxy+by2+2gx+2fy+c=0 intersect on the y-axis then

2fgh=bg2+ch2

bg2=ch2

abc=2fgh