The value of λ for which the lines joining the points of intersection of curves C1 and C2 to the origin are equally inclined to the axis of x, where C2: λ x2+3 y2-2 λ x y+9 x=0 and C1: 3 x2-4 y2+8 x y-3 x=0, is

Question

The value of λ for which the lines joining the points of intersection of curves C1 and C2 to the origin are equally inclined to the axis of x, where C2: λ x2+3 y2-2 λ x y+9 x=0 and C1: 3 x2-4 y2+8 x y-3 x=0, is (A)  λ=(4/3) (B) λ=12 (C) λ=1 (D) λ=(5/6)

Tardigrade Admin · Accepted Answer

To find the equation of pair of lines through origin we eliminate the terms containing

x

from the equations of

C_{1}

and

C_{2}

.
Thus,

C_{1} + 3 C_{2} = 0

\Rightarrow (λ + 9) x^{2} + (24 - 2 λ) x y - 9 y^{2} = 0

These lines are equally inclined to axes if

24 - 2 λ = 0

\Rightarrow λ = 12

Q. The value of $λ$ for which the lines joining the points of intersection of curves $C_{1}$ and $C_{2}$ to the origin are equally inclined to the axis of $x$ , where $C_{2}$ : $λ x^{2} + 3 y^{2} - 2 λ x y + 9 x = 0$ and $C_{1} : 3 x^{2} - 4 y^{2} + 8 x y - 3 x = 0$ , is

$λ = \frac{4}{3}$

$λ = 12$

$λ = 1$

$λ = \frac{5}{6}$

Q. The value of λ for which the lines joining the points of intersection of curves C1​ and C2​ to the origin are equally inclined to the axis of x, where C2​ : λx2+3y2−2λxy+9x=0 and C1​:3x2−4y2+8xy−3x=0, is

λ=34​

λ=12

λ=1

λ=65​

To find the equation of pair of lines through origin we eliminate the terms containing x from the equations of C1​ and C2​. Thus, C1​+3C2​=0 ⇒(λ+9)x2+(24−2λ)xy−9y2=0 These lines are equally inclined to axes if 24−2λ=0 ⇒λ=12

Q. The value of $λ$ for which the lines joining the points of intersection of curves $C_{1}$ and $C_{2}$ to the origin are equally inclined to the axis of $x$ , where $C_{2}$ : $λ x^{2} + 3 y^{2} - 2 λ x y + 9 x = 0$ and $C_{1} : 3 x^{2} - 4 y^{2} + 8 x y - 3 x = 0$ , is

$λ = \frac{4}{3}$

$λ = 12$

$λ = 1$

$λ = \frac{5}{6}$