a curve is defined in the x-y plane by y=(x-1)(ax^2+bx+x) where a, b and c are constants. the curve touches the x-axis at x=1 and the line y=3x+6. a) find the values of a, b and c.
Fix the typo (ax^2+bx+x ?), and is the curve tangent to the line?
Solution
23
To find the values of a, b, and c, we need to use the given information about the curve.
1) The curve touches the x-axis at x=1:
When a curve touches or crosses the x-axis, it means that the value of y becomes zero at that particular value of x. Therefore, we can substitute x=1 into the equation y = (x-1)(ax^2 + bx + c) and solve for y=0.
0 = (1-1)(a(1)^2 + b(1) + c)
0 = (0)(a + b + c)
0 = 0
From this equation, we can see that the value inside the parentheses (a + b + c) can be any real number since multiplying it by zero will always result in zero. Therefore, we can't determine the values of a, b, and c from this condition.
2) The curve touches the line y=3x+6:
For the curve to touch the line y=3x+6, the equation of the curve should have the same slope as the line in order for it to touch at one point only.
The slope of the line y=3x+6 is 3. Since the curve touches the line at x=1, we can find the slope of the curve at that point by taking the derivative of the equation y = (x-1)(ax^2 + bx + c) and evaluating it at x=1.
Taking the derivative:
dy/dx = [(1-1)(ax^2 + bx + c) + (x-1)(2ax + b)]
Evaluating at x=1:
dy/dx = [0 + (1-1)(2a + b)]
dy/dx = (2a + b)
We know that the slope of the curve at x=1 should be equal to the slope of the line, which is 3.
Therefore, we have the equation:
2a + b = 3 ----(1)
So, we have one equation (1) with two variables (a and b). We need to find another equation to solve for the values of a, b, and c.
Let's look back at the equation of the curve:
y = (x-1)(ax^2 + bx + c)
We know that the curve touches the x-axis at x=1, which means that the point (1,0) lies on the curve. We can substitute these values into the equation to get another equation:
0 = (1-1)(a(1)^2 + b(1) + c)
0 = (0)(a + b + c)
0 = 0
As we can see, this equation doesn't provide any additional information because it is the same equation we obtained earlier when we touched the x-axis condition.
From the information given, we can only determine the value of the slope of the curve at x=1, which gives us one equation (1) with two variables (a and b). Without any other given conditions or equations, it is not possible to find the exact values of a, b, and c.