Find the equation of the line passing through (1, 2) and making an angle of 30° with the y-axis.
Understand the Problem
The question is asking to find the equation of a line that passes through the point (1, 2) and makes an angle of 30° with the y-axis. This involves using the concept of slopes and coordinate geometry.
Answer
The equation of the line is $$ y = \sqrt{3}x + (2 - \sqrt{3}) $$.
Answer for screen readers
The equation of the line is $$ y = \sqrt{3}x + (2 - \sqrt{3}) $$.
Steps to Solve
- Determine the Slope from the Angle
The slope ($m$) of a line that makes an angle $\theta$ with the y-axis can be calculated using the tangent function: $$ m = \tan(90° - \theta) $$
Since the line makes a 30° angle with the y-axis, we can calculate: $$ m = \tan(90° - 30°) = \tan(60°) $$
- Calculate the Tangent of 60°
Using the known value: $$ \tan(60°) = \sqrt{3} $$
So, the slope of our line is: $$ m = \sqrt{3} $$
- Use the Point-Slope Form of the Line Equation
The point-slope form of the equation of a line is given as: $$ y - y_1 = m(x - x_1) $$
Substituting the point (1, 2) and the slope $\sqrt{3}$ into the formula gives: $$ y - 2 = \sqrt{3}(x - 1) $$
- Rearrange to Slope-Intercept Form
To rearrange the equation into slope-intercept form ($y = mx + b$): $$ y - 2 = \sqrt{3}x - \sqrt{3} $$
So, $$ y = \sqrt{3}x - \sqrt{3} + 2 $$
- Simplify the Equation
Combine terms to get the final equation: $$ y = \sqrt{3}x + (2 - \sqrt{3}) $$
The equation of the line is $$ y = \sqrt{3}x + (2 - \sqrt{3}) $$.
More Information
- The line passes through the point (1, 2) and is inclined at 30° to the y-axis, hence its slope corresponds to the angle.
- This line will intersect the y-axis at the point $2 - \sqrt{3}$.
Tips
- Confusing the angle with respect to the x-axis instead of the y-axis, which would lead to incorrect slope calculation.
- Forgetting to rearrange the final equation into the standard form $y = mx + b$.
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