Question

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

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

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

$x^2(1-m^2)+2mxy=0$

which represents the point of lines between which

the angler is ${45}^{\circ }$

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

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

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

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

$m^2+2m-1=0$ $m^2-2m-1=0$

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

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

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

$\therefore Opt:\left[C\right]$ is the answer