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  • Set of values of $m$ for which a chord of slope m of the circle touches the par

Set of values of $m$ for which a chord of slope m of the circle x^2+y^2=4 touches the parabola y^2=4 x, is

Option: 1

\left(-\infty,-\sqrt{\frac{\sqrt{2}-1}{2}}\right) \cup\left(\sqrt{\frac{\sqrt{2}-1}{2}}, \infty\right)


Option: 2

(-\infty,-1) \cup(1, \infty)


Option: 3

(-1,1)


Option: 4

R


Answers (1)

(a) The equation of tangent of slope m to the parabola

                  y^2=4 x \text { is } y=m x+\frac{1}{m}

This will be a chord of the circle x^2+y^2=4, if length of the perpendicular from then centre (0,0) is less than the radius i.e., \left|\frac{1}{m \sqrt{m^2+1}}\right|<2

\begin{aligned} & \Rightarrow 4 m^4+4 m^2-1>0 \\ \\& \Rightarrow\left(m^2-\frac{\sqrt{2}-1}{2}\right)\left(m^2+\frac{\sqrt{2}+1}{2}\right)>0 \\ \\& \Rightarrow\left(m^2-\frac{\sqrt{2}-1}{2}\right)>0 \end{aligned}

\begin{aligned} & \Rightarrow\left(m^2-\sqrt{\frac{\sqrt{2}-1}{2}}\right)\left(m+\sqrt{\frac{\sqrt{2}-1}{2}}\right)>0 \\ \\& \Rightarrow \quad m \in\left(-\infty,-\sqrt{\frac{\sqrt{2}-1}{2}}\right) \cup\left(\sqrt{\frac{\sqrt{2}-1}{2}}, \infty\right) \end{aligned}

Posted by

Ramraj Saini

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