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If the chord $y=mx+1$ of the circle $x^2+y^2=1$ substance on angle of measures $45^{\circ}$ at the major segment of the circle then value of m is

$\begin{array}{1 1}(A)\;2 \pm \sqrt 2 \\(B)\;-2 \pm \sqrt 2 \\(C)\;-1 \pm \sqrt 2 \\(D)\;none\;of\;these \end{array}$

1 Answer

Equation of circle $x^2+y^2=1$
=> $x^2+y^2=(y-mx)^2$
=> $x^2(1-m^2)+2mxy=0$
Which represents the pair of lines between which the angle is $45^{\circ}$
$\tan 45=\pm \large\frac{2 \sqrt m^2-0}{1-m^2}$
$\qquad= \large\frac{\pm 2m}{1-m^2}$
=> $1=m^2=\pm 2m$
=> $m^2 \pm 2m-1=0$
=>$m=\large\frac{-2 \pm \sqrt {4+4}}{2}$
$\qquad= \large\frac{-2 \pm 2 \sqrt 2}{2}$
$\qquad= -1 \pm \sqrt 2$
Hence C is the correct answer.
answered Apr 8, 2014 by meena.p

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