Question

If the chord $y=mx+!$ of the circle $x^2+y^2=1$ subtend an angle of measure ${45}^o$ at the major segment of the circle then value of $m$ is

• A. $2\pm \sqrt{2}$
• B. $-2\pm \sqrt{2}$
• C. $-1\pm \sqrt{2}$
• D. none of these

Answer 1

The correct option is C $-1\pm \sqrt{2}$

Equation of circle $x^2+y^2=1$

$\Rightarrow x^2+y^2=(y-mx{)}^2$

$\Rightarrow x^2=m^2{x}^2-2mxy$

$\Rightarrow x^2(1-m^2{)}_{2}mxy=0$

Which represents the pair of lines between which the angle is ${45}^o$

$\tan 45=\frac{\pm 2\sqrt{m^2}-0}{1-m^2}$

$\Rightarrow 1=\frac{\pm 2m}{1-m^2}$

$\Rightarrow 1-m^2=\pm 2m$

$\Rightarrow m^2\pm 2m-1=0$

$\Rightarrow m=\frac{-2\pm \sqrt{4+4}}{2}$

$\Rightarrow m=\frac{-2\pm 2\sqrt{2}}{2}$

$\Rightarrow m=-1\pm \sqrt{2}$