First slide

Definition of a circle

Question

If the chord $y = mx + 1$ of the circles $x^{2} + y^{2} = 1$ subtends an angle of 45° at the major segment of the circle, then the value of m is

Moderate

A

2
B

-2
C

-1
D

none of these

Solution

The given circle is
$x^{2} + y^{2} = 1$
$C \equiv (0, 0), radius = 1$
and chord is. $y = mx + 1$
$\cos 45^{\circ} = \frac{CP}{CR}$
Perpendicular distance from
(0, 0) to the chord y = mx +1,
$CP = \frac{1}{\sqrt{m^{2} + 1}}, (CR = Radius = 1)$
or $\cos 45^{\circ} = \frac{1 / \sqrt{m^{2} + 1}}{1}$
or $\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{m^{2} + 1}}$
or $m^{2} + 1 = 2$
or $m = \pm 1$

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