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The given figure represents the lines y = x + 1 and y = 3x1.\sqrt{3}x - 1. Write down the angles which the lines make with the positive direction of the x-axis. Hence, determine θ.

The given figure represents the lines y = x + 1 and y = √3x - 1. Write down the angles which the lines make with the positive direction of the x-axis. Hence, determine θ. Equation of a Straight Line, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Straight Line Eq

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Answer

Given,
y = x + 1 and y = 3x1\sqrt{3}x - 1.

Comparing equations with y = mx + c we get,

m1 = 1 and m2 = 3\sqrt{3}.

Let the first line make angle θ1 and second make θ2 with positive direction of x-axis.

The inclination that y = x + 1 makes is,
⇒ m1 = tan θ1 = 1
⇒ tan θ1 = 1 = tan 45°
⇒ tan θ1 = tan 45°
⇒ θ1 = 45°.

The inclination that y = 3x\sqrt{3}x - 1 makes is,
⇒ m2 = tan θ2 = 3\sqrt{3}
⇒ tan θ2 = 3\sqrt{3} = tan 60°
⇒ tan θ2 = tan 60°
⇒ θ2 = 60°.

From graph we get,

The given figure represents the lines y = x + 1 and y = √3x - 1. Write down the angles which the lines make with the positive direction of the x-axis. Hence, determine θ. Equation of a Straight Line, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

60° is the exterior angle. We know that,

Exterior angle = Sum of two opposite interior angles.

∴ 60° = θ + 45°
⇒ θ = 60° - 45°
⇒ θ = 15°

Hence, y = x + 1 makes 45° and y = 3x1\sqrt{3}x - 1 makes 60° with the x-axis. The value of θ = 15°.

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