Q77 / 200 · WBJEE 2014 · Mathematics

The straight lines $x+y=0,5 x+y=4$ and $x+5 y=4$ form

  • Correct answer
    A
    an isosceles triangle
  • B
    an equilateral triangle
  • C
    a scalene triangle
  • D
    a right angled triangle

Solution

Given lines are $$ \begin{array}{r} x+y=0 \\ 5 x+y=4 \end{array} $$ and $$ x+5 y=4 $$ On solving above equations pairwise, we get $$ \begin{array}{l} A(1,-1), B\left(\frac{2}{3}, \frac{2}{3}\right), C(-1,1) \\ Now, A B=\sqrt{\left(\frac{2}{3}-1\right)^{2}+\left(\frac{2}{3}+1\right)^{2}} \end{array} $$ $$ \begin{array}{l} =\sqrt{\frac{1}{9}+\frac{25}{9}}=\sqrt{\frac{26}{9}}=\frac{\sqrt{26}}{3} \\ \begin{aligned} B C &=\sqrt{\left(-1-\frac{2}{3}\right)^{2}+\left(1-\frac{2}{3}\right)^{2}} \\ &=\sqrt{\frac{25}{9}+\frac{1}{9}}=\sqrt{\frac{26}{9}}=\frac{\sqrt{26}}{3} \\ C A=\sqrt{(1+1)^{2}+(-1-1)^{2}} \end{aligned} \end{array} $$ $$ =\sqrt{4+4}=2 \sqrt{2} $$ Here, we see that $A B=B C$ Hence, given lines form an isosceles triangle.

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

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