Question 17 of 70intermediate🔧 ApplyVery Short2 marks

In Exercises 1 to 8, find the equation of the line which satisfy the given conditions: Intersecting the yy-axis at a distance of 2 units above the origin and making an angle of 30∘30^\circ with positive direction of the xx-axis.

Correct Answer

The line intersects the y-axis at (0, 2) since it is 2 units above the origin.

The line makes an angle of 30° with the positive direction of the x-axis. Therefore, the slope of the line is: m = tan 30° = 1/3\sqrt{3}

Using the point-slope form of the equation of a line passing through point (x₁, y₁) with slope m: y - y₁ = m(x - x₁)

Substituting the point (0, 2) and slope m = 1/3\sqrt{3}: y - 2 = (1/3\sqrt{3})(x - 0) y - 2 = x/3\sqrt{3}

Therefore, the equation of the line is: y = x/3\sqrt{3} + 2

Or in standard form: x - 3\sqrt{3}y + 23\sqrt{3} = 0

Exercise: EXERCISE 9.2 | Q: 6 | (Page 13)
For More Understanding

Explanation

This question tests the student's understanding of finding the equation of a line when given a point on the line and the angle it makes with the x-axis. The key steps are: (1) identifying the point of intersection on the y-axis, (2) converting the angle to slope using m = tan θ, and (3) applying the point-slope form. The textbook context shows similar methodology in Example 12 where slope is found as tan θ and point-slope form is used.

Solution Steps

  1. Step 1: Identify the point of intersection on y-axis: (0, 2)

  2. Step 2: Calculate slope: m = tan 30° = 1/3\sqrt{3}

  3. Step 3: Apply point-slope form: y - 2 = (1/3\sqrt{3})(x - 0)

  4. Step 4: Simplify to get: y = x/3\sqrt{3} + 2 or x - 3\sqrt{3}y + 23\sqrt{3} = 0

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