f two tangents from point (3, –2) to the circle x2 + y2 = 1 cuts x-axis at A(x1, 0) and B(x2, 0), then (x1 + x2) is equal to
Question
f two tangents from point (3, –2) to the circle x2 + y2 = 1 cuts x-axis at A(x1, 0) and B(x2, 0), then (x1 + x2) is equal to
Solution
The problem involves a circle and tangents from an external point. Here are the steps to solve it:
Step 1: Write down the given information. The equation of the circle is x² + y² = 1, and the external point is (3, -2).
Step 2: The equation of the tangent line from an external point (x1, y1) to the circle x² + y² = r² is given by the formula: xx1 + yy1 = r². So, the equation of the tangent line from (3, -2) to the circle x² + y² = 1 is: 3x - 2y = 1.
Step 3: Since the tangent line cuts the x-axis at A(x1, 0) and B(x2, 0), we can substitute y = 0 into the equation of the tangent line to find the x-coordinates of A and B. Doing this gives us: 3x = 1, so x = 1/3.
Step 4: Therefore, both x1 and x2 are equal to 1/3, so x1 + x2 = 1/3 + 1/3 = 2/3.
So, the sum of the x-coordinates of the points where the tangents cut the x-axis is 2/3.
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