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The curve C has equation $y = \frac{1}{3}x^2 + 8$\nThe line L has equation $y = 3x + k$, where k is a positive constant.\n\n(a) Sketch C and L on separate diagrams, showing the coordinates of the points at which C and L cut the axes.\n\nGiven that line L is a tangent to C,\n(b) find the value of k. - Edexcel - A-Level Maths Pure - Question 11 - 2014 - Paper 2
Question 11
The curve C has equation $y = \frac{1}{3}x^2 + 8$\nThe line L has equation $y = 3x + k$, where k is a positive constant.\n\n(a) Sketch C and L on separate diagrams, ... show full transcript
Worked Solution & Example Answer:The curve C has equation $y = \frac{1}{3}x^2 + 8$\nThe line L has equation $y = 3x + k$, where k is a positive constant.\n\n(a) Sketch C and L on separate diagrams, showing the coordinates of the points at which C and L cut the axes.\n\nGiven that line L is a tangent to C,\n(b) find the value of k. - Edexcel - A-Level Maths Pure - Question 11 - 2014 - Paper 2
Step 1
Sketch C and L
Answer
To sketch the curve C, first determine the vertex at point (0, 8), as the graph opens upward. It intercepts the y-axis at (0, 8). For x-intercepts, set ( \frac{1}{3}x^2 + 8 = 0 ) which gives no real solutions since the graph does not intersect the x-axis.
For line L, with equation ( y = 3x + k ):
- The y-intercept is at (0, k).
- The x-intercept can be found by setting y to zero: ( 0 = 3x + k \Rightarrow x = -\frac{k}{3} ). This gives the coordinates of line L as (0, k) and (-\frac{k}{3}, 0).
Both the curve and the line should be sketched clearly in separate diagrams showing the intercepts and respective shapes.
Step 2
find the value of k
Answer
Since line L is tangent to the curve C, the two equations must intersect at exactly one point. Set the two equations equal to each other for intersection: [ \frac{1}{3}x^2 + 8 = 3x + k ] Rearranging this gives: [ \frac{1}{3}x^2 - 3x + (8 - k) = 0 ] This is a quadratic equation, and for it to have one solution, the discriminant must equal zero: [ D = b^2 - 4ac = 0\n\Rightarrow (-3)^2 - 4(\frac{1}{3})(8 - k) = 0 ] Solving this gives: [ 9 - \frac{4}{3}(8 - k) = 0\n\Rightarrow 9 = \frac{32 - 4k}{3}\n\Rightarrow 27 = 32 - 4k\n\Rightarrow 4k = 32 - 27\n\Rightarrow 4k = 5\n\Rightarrow k = \frac{5}{4} ] Therefore, the value of k is ( \frac{5}{4} ).