The curve y = 7 - \(\frac{6}{x}\) and the line y + 2x - 3 = 0 intersect at two point. Finf the;
(a) coordinates of the two points
(b) equation of the perpendicular bisector of the line joining the two points
(a) coordinates of the two points
(b) equation of the perpendicular bisector of the line joining the two points
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Correct Answer: Option
Explanation:
(a) 7 - \(\frac{6}{x} = 3 - 2x\)
Simplifying; \(x^2 + 2x - 2 = 0\)
x = 1 or x = -3
Substituting for x; y = 3 - 2(1) = 3 - 2 = 1 or y = 3 - 2(-3) = 3 + 6 = 9
The coordinate of the two points are (x y) = (1, 1), (-3, 9)
(b) (\(\frac{1 - 3}{2}, \frac{1 + 9}{2}\)) = (-1, 5)
The gradient of the point of intersection ; \(\frac{9 - 1}{-3 -1} = \frac{8}{-4}\) = -2
The gradient of the perpendicular bisector; \(\frac{1}{2}\)
Thus, the equation of the perpendicular bisector; y - 5 = \(\frac{1}{2}\) (x + 1)
Therefore, 2y - x - 11 = 0
(a) 7 - \(\frac{6}{x} = 3 - 2x\)
Simplifying; \(x^2 + 2x - 2 = 0\)
x = 1 or x = -3
Substituting for x; y = 3 - 2(1) = 3 - 2 = 1 or y = 3 - 2(-3) = 3 + 6 = 9
The coordinate of the two points are (x y) = (1, 1), (-3, 9)
(b) (\(\frac{1 - 3}{2}, \frac{1 + 9}{2}\)) = (-1, 5)
The gradient of the point of intersection ; \(\frac{9 - 1}{-3 -1} = \frac{8}{-4}\) = -2
The gradient of the perpendicular bisector; \(\frac{1}{2}\)
Thus, the equation of the perpendicular bisector; y - 5 = \(\frac{1}{2}\) (x + 1)
Therefore, 2y - x - 11 = 0