Question

If the chord $y=mx+1$ of circle $x^2+y^2=1$ subtends an angle of measure ${45}^{\circ }$ 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

Answer: (C)

$-1\pm \sqrt{2}$

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

Joint equation of the curve through the intersection of line and circle be given

$x^2+y^2=(y-mx{)}^2$ (By homogenizing the equation)

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

Now $\tan \theta =\pm \frac{2\sqrt{h^2-ab}}{a+b}$ where $\{\begin{array}{c}a=1-m^2\\ h=m,b=0\end{array}$

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

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

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

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

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