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A sketch of part of the curve C with equation y = 20 - 4x - 18/x, x > 0 is shown in Figure 3 - Edexcel - A-Level Maths Pure - Question 1 - 2013 - Paper 2
Question 1
A sketch of part of the curve C with equation y = 20 - 4x - 18/x, x > 0 is shown in Figure 3. Point A lies on C and has an x coordinate equal to 2. (a) Show tha... show full transcript
Worked Solution & Example Answer:A sketch of part of the curve C with equation y = 20 - 4x - 18/x, x > 0 is shown in Figure 3 - Edexcel - A-Level Maths Pure - Question 1 - 2013 - Paper 2
Step 1
Show that the equation of the normal to C at A is y = -2x + 7
Answer
To find the equation of the normal to the curve at point A, we first need to determine the slope of the tangent at A. We start by substituting the x-coordinate of A into the equation of the curve:
- Substitute x = 2 into the equation:
y = 20 - 4(2) - \frac{18}{2} = 20 - 8 - 9 = 3. Thus, point A is (2, 3).
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Next, find the derivative to determine the slope of the tangent:
\frac{dy}{dx} = -4 + \frac{18}{x^2}.
Substituting x = 2: \frac{dy}{dx} = -4 + \frac{18}{2^2} = -4 + \frac{18}{4} = -4 + 4.5 = 0.5.
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The slope of the normal line is the negative reciprocal of the slope of the tangent: \text{slope of normal} = -\frac{1}{0.5} = -2.
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Using point-slope form of the line: y - y_1 = m(x - x_1), where (x_1, y_1) = (2, 3) and m = -2: y - 3 = -2(x - 2) y - 3 = -2x + 4 y = -2x + 7. Therefore, the equation of the normal line is confirmed as y = -2x + 7.
Step 2
Use algebra to find the coordinates of B
Answer
To find the coordinates of point B, we need to find the intersection of the normal line and the curve C.
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Set the equation of the normal line equal to the equation of the curve:
-2x + 7 = 20 - 4x - \frac{18}{x}.
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Rearranging gives: 2x + 7 = 20 - 4x - \frac{18}{x}, 6x + 7 = 20 - \frac{18}{x}.
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Multiply through by x to eliminate the fraction (assuming x ≠ 0): 6x^2 + 7x = 20x - 18.
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Rearranging leads to: 6x^2 - 13x + 18 = 0.
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Factoring the quadratic gives: (3x - 2)(2x - 9) = 0. This gives: x = \frac{2}{3} \quad or \quad x = \frac{9}{2}. Since we are interested in B (the other intersection), we use x = \frac{9}{2}.
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Substitute x back into the equation of the curve to find y:
y = 20 - 4(\frac{9}{2}) - \frac{18}{\frac{9}{2}} = 20 - 18 + -4 = -2.
- Thus, the coordinates of point B are ( (\frac{9}{2}, -2) ).