Question 24 of 70intermediate🔧 ApplyShort Answer4 marks

Find the equation of the line through the point (0,2)(0, 2) making an angle 2π3\frac{2\pi}{3} with the positive xx-axis. Also, find the equation of line parallel to it and crossing the yy-axis at a distance of 2 units below the origin.

Correct Answer

(i) 3\sqrt{3}x + y - 2 = 0

(ii) 3\sqrt{3}x + y + 2 = 0

Exercise: EXERCISE 9.2 | Q: 13 | (Page 14)
For More Understanding

Explanation

This question tests the application of the point-slope form of a line equation. The slope is derived from the given angle using the tangent function. For the parallel line, the same slope is used with a different y-intercept.

Solution Steps

  1. Step 1: Find the slope of the line making angle 2π/3 with positive x-axis.

    Slope m = tan(2π/3) = tan(120°) = -3\sqrt{3}

  2. Step 2: Use point-slope form for line through (0, 2) with slope -3\sqrt{3}.

    Using y - y₀ = m(x - x₀) y - 2 = -3\sqrt{3}(x - 0) y - 2 = -3\sqrt{3}x 3\sqrt{3}x + y - 2 = 0

  3. Step 3: For parallel line, use same slope m = -3\sqrt{3}.

    The line crosses y-axis at 2 units below origin, so y-intercept = -2. Using point-slope form through (0, -2): y - (-2) = -3\sqrt{3}(x - 0) y + 2 = -3\sqrt{3}x 3\sqrt{3}x + y + 2 = 0

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