Question

If the equations of the hypotenuse and a side of a right-angled isosceles triangle be $x+my=1$ and $x=k$ respectively then

• A. $m=1$
• B. $m=k$
• C. $m=-1$
• D. $m+k=0$

Answer 1

Answer: (A), (C)

The correct options are
A $m=1$
C $m=-1$

Given sides

$x+my=1$----(1)

and $x=k$---(2)

On comparing both eq with $y=mx+c$ we get

$m_1=-\frac{1}{m}$ and $m_2=1$

And The third sides of traingle is perpendicular to $x-k=0$ So slope of third line be $m_2{m}_3=-1$

$m_3=-1$

The given triangle is isosceles triangle so the angle between line (1) and (2) is equal to angle between line (1) and (3)

$\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|=\left|\frac{m_1-m_3}{1+m_1{m}_3}\right|$

$\left|\frac{-\frac{1}{m}-1}{1-\frac{1}{m}}\right|=\left|\frac{-\frac{1}{m}+1}{1+\frac{1}{m}}\right|$

$\left|\frac{-1-m}{m-1}\right|=\left|\frac{-1+m}{m+1}\right|$

$|-(1+m{)}^2|=\left|\right(m-1{)}^2|$

$|-1-m^2-2m|=|m^2+1-2m|$

$2m^2=|-2|$

$m^2=1$

$m=\pm 1$