Question

One vertex of an equilateral triangle is $(2,3)$ and the equation of one side is $x-y+5=0$, then the equations to the other sides are

• A. $y-3=-(2\pm \sqrt{3})(x-2)$
• B. $y-3=(\sqrt{2}\pm 1)(x-2)$
• C. $y-3=(\sqrt{3}\pm 1)(x-2)$
• D. $y-3=(\sqrt{5}\pm 1)(x-2)$

Answer 1

The correct option is A $y-3=-(2\pm \sqrt{3})(x-2)$

Equation of one side of equilateral triangle is $x-y+5=0$

Let the slope of the other two lines be $m_1$, $m_2$ and $m_3=1$ ....($m_3$ is of given line)

Let the angle between any two lines is $\theta =\frac{\pi }{3}$
$\tan \theta =\sqrt{3}=\frac{m_1-1}{1+m_1}=\frac{1-m_2}{1+m_2}$

Solving these, we get

$m_1=-(2+\sqrt{3})$ and $m_2=-(2-\sqrt{3})$ .
$\therefore$ the equations of the lines passing through $(2,3)$ and with the slopes $-(2\pm \sqrt{3})$ is given by,

$y-3=-(2\pm \sqrt{3})(x-2)$.

Hence, option A is correct answer.