Photo AI
The curve C has equation $y = \frac{1}{3}x^2 + 8$ - Edexcel - A-Level Maths Pure - Question 11 - 2014 - Paper 2
Question 11
The curve C has equation $y = \frac{1}{3}x^2 + 8$. The line L has equation $y = 3x + k$, where $k$ is a positive constant. (a) Sketch C and L on separate diagrams,... show full transcript
Worked Solution & Example Answer:The curve C has equation $y = \frac{1}{3}x^2 + 8$ - Edexcel - A-Level Maths Pure - Question 11 - 2014 - Paper 2
Step 1
Sketch C and L on separate diagrams
Answer
To sketch the curve C, we start by plotting the vertex at the point (0, 8) since the vertex form for the parabola is .
-
Identify the intercepts for C:
- The y-intercept occurs when :
- , hence .
- The x-intercepts are found by setting :
- Solve , which gives no real x-intercepts because is positive.
- The y-intercept occurs when :
-
Shape of C:
- The graph is an upward-opening parabola, symmetric about the y-axis.
-
Sketch line L:
- For the line , it will also cross the y-axis at the point .
- The slope is positive, indicating the line will rise as x increases.
-
Identify points of intersection:
- Line L will cut the y-axis at , and its x-intercept can be found by setting :
- , hence the intercept .
- Line L will cut the y-axis at , and its x-intercept can be found by setting :
Step 2
find the value of k
Answer
Since line L is tangent to curve C, at the point of tangency, both the y-values and the slopes of the two equations must be equal.
- Equate curves:
- Set the equations equal:
- Rearranging gives:
- Condition for tangency:
- For tangency, the discriminant must be zero:
- Here, .
- Calculate the discriminant:
- Solving the equation gives: Thus, the value of k is (1.25).