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 value of m is

• A. 2
• B. – 2
• C. $\pm$ 1
• D. None of these

Answer 1

Answer: (C)

$\pm$ 1

Answer 2

Given circle is $x^{2} + y^{2} = 1,$ C (0,0) and radius = 1 and chord is $y = mx + 1$

$\cos 45^{o} = \frac{CP}{CR}$; CP = Perpendicular distance from (0, 0) to chord $y = mx + 1$

$CP = \frac{1}{\sqrt{m^{2} + 1}}$ (CR = radius =1)

$\cos 45^{o} = \frac{\frac{1}{\sqrt{m^{2} + 1}}}{1} \Rightarrow \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{m^{2} + 1}}$

⇒ $m^{2} + 1 = 2 \Rightarrow m = \pm 1.$