A line passing through the point P(a,θ) makes an acute angle α with the positive x-axis. Let this line be rotated about the point P through an angle 2α in the clockwise direction. If in the new position, the slope of the line is 2−3 and its distance from the origin is 21, then the value of 3a2tan2α−23 is:
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Step-by-Step Solution
Step 1: Determine the initial and final angles of the line
The initial line makes an angle α with the positive x -axis, so its slope is m1=tanα. When the line is rotated clockwise by 2α, its new angle with the positive x -axis becomes α−2α=2α. Therefore, the slope of the new line is m2=tan(2α).
Step 2: Find the value of α
We are given that the slope of the new line is 2−3. We know that tan15∘=2−3. Therefore, 2α=15∘, which implies α=30∘.
Step 3: Determine the equation of the new line
The new line passes through the point P(a,θ) and has a slope of 2−3. Using the point-slope form of a line, y−y1=m(x−x1), we get the equation of the line.
Step 4: Use the distance from the origin to find 'a'
The distance of a line Ax+By+C=0 from the origin is given by A2+B2∣C∣. Substituting the values from our line equation, we can simplify the denominator. Note that 8−43=2(4−23)=2(3−1).
Step 5: Evaluate the expression
We need to find the value of 3a2tan2α−23. We already found α=30∘. Substituting this value, we get tan30∘=31. The expression simplifies to a2−23.