Question

If the points of intersection of curves $C_1=\lambda x^2+4y^2-2xy-9x+3and{C}_2=2x^2+3y^2-4xy+3x-1$ subtends a right angle at origin then the value of $\lambda$ is

• A. $19$
• B. $9$
• C. $-19$
• D. $-9$

Answer 1

Answer: (C)

$-19$

Given

$C_1:\lambda x^2+4y^2-2xy-9x+3=0$

$C_2:2x^2+3y^2-4xy+3x-1=0\Rightarrow 2x^2+3y^2-4xy-1=-3x$-----(1)

Putting (1) in $C_1$

$\lambda x^2+4y^2-2xy+3(2x^2+3y^2-4xy-1)+3=0$

$\lambda x^2+4y^2-2xy+6x^2+9y^2-12xy-3+3=0$

$(\lambda +6)x^2+13y^2-14xy=0$

Above equation has a condition of perpendicularity

Hence $\lambda +6+13=0\Rightarrow \lambda =-19$