Final answer:
The equations of two lines passing through the point (7,9) and making an angle of 60° with the line x−√3y−2√3=0 are x=7, and x+√3y=7+9√3. These are derived using formulas for the angle between two lines and the equation of a line through a point.
Step-by-step explanation:
To solve this question, we need to use the concept of the angle between two lines and the equation of a line passing through a point. In general, for an angle of θ between two lines, the slope of the lines can be obtained from the formula tan(θ) = |(m1 - m2) / (1 + m1*m2)|, where m1 and m2 are the slopes of the two lines.
From the line equation x−√3y−2√3=0, we obtain the slope m1 = √3. The slope of the second line passing through the point (7,9) and making an angle of 60° with the first line can be obtained by rearranging the formula, giving you m2 = -m1 or m1*m2 = -tan(60°).
These relationships can be used to form two new equations with two different slopes: m1 = -√3 and m2 = √3. Using the point (7,9) and the new slopes, we can establish two equations for the lines passing through the point (7,9) and making an angle of 60° with the original line: x = 7 (i.e., a vertical line passing through (7,9)) and x+√3y = 7+9√3.
So the correct option is a: x=7, x+√3y=7+9√3.
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