# If the chord  $\dpi{100} y=mx+1$ of the circle $\dpi{100} x^{2}+y^{2}=1$   subtends an angle of measure 45° at the major segment of the circle then value of m is Option 1) $2\pm \sqrt{2}$ Option 2) $-2\pm \sqrt{2}$ Option 3) $-1\pm \sqrt{2}$ Option 4) none of these

As we learnt in

Condition of tangency -

$c^{2}=a^{2}\; (1+m^{2})$

- wherein

If  $y=mx+c$  is a tangent to the circle $x^{2}+y^{2}=a^{2}$

Equation of circle x2+y2=1

Now, y=mx+1

$\\ So, \: \: x^{2}+y^{2}=(y-mx)^{2} \\ \\ x^{2}\left ( 1-m^{2} \right )+2mxy=0$

which represents pair of lines with $angle=45^{\circ}$

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

On solving  $m=\frac{-2\pm 2\sqrt{2}}{2} \: =-1\pm \sqrt{2}$

Option 1)

$2\pm \sqrt{2}$

This option is incorrect

Option 2)

$-2\pm \sqrt{2}$

This option is incorrect

Option 3)

$-1\pm \sqrt{2}$

This option is correct

Option 4)

none of these

This option is incorrect

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