Q60 / 160 · TS EAMCET 2003 · Mathematics

The area (in sq units) of the quadrilateral formed by two pairs of lines $\lambda^2 x^2-m^2 y^2-n(\lambda x+m y)=0 \quad$ and $\lambda^2 x^2-m^2 y^2+n(\lambda x+m y)=0$ is:

  • Correct answer
    A
    $\frac{n^2}{2|\lambda m|}$
  • B
    $\frac{n^2}{|\lambda m|}$
  • C
    $\frac{n}{2|\lambda m|}$
  • D
    $\frac{n^2}{4|\lambda m|}$

Solution

Equation of lines are $\lambda x+m y=0, \lambda x-m y-n=0$ and $\quad \lambda x-m y=0, \lambda x+m y+n=0$ or $\quad \lambda x+m y=0, \lambda x+m y+n=0$ and $\quad \lambda x-m y=0, \lambda x-\lambda m-n=0$ Area $=\left|\frac{\left(c_1-d_1\right)\left(c_2-d_2\right)}{a_1 b_2-a_2 b_1}\right|$ $=\left|\frac{(0-n)(0+n)}{-\lambda m-\lambda m}\right|$ $=\frac{n^2}{2|\lambda m|}$ sq units

Hence, the correct answer is option: 'A'.

Relevant Questions