Question

If the line $x^2+2xy-35y^2-4x+44y-12=0$ and $5x+\lambda y-8=0$ are concurrent then the value of $\lambda$ is

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

The correct option is C $2$

Point of intersection of pairs of line represented by $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ is $(\frac{hf-bg}{ab-h^2},\frac{gh-af}{ab-h^2})$

for the lines

$x^2+2xy-35y^2--12=0$

$a=1,h=1,b=-35,g=-2,f=22,c=-12$

$\frac{hf-bg}{ab-h^2}=\frac{(-2)\left(1\right)-\left(1\right)22}{-(-35)-1^2}=\frac{-24}{-36}=\frac{2}{7}$

point of intersection $\equiv (\frac{4}{3},\frac{2}{3})$

This must satisfy $5x+\lambda y-8=0$

$\Rightarrow 5\left(\frac{4}{3}\right)+\lambda \left(\frac{2}{3}\right)-8=0$

$\Rightarrow 20+2\lambda -24=0$

$2\lambda =4$

$\lambda 2$