The tangent to the curve 2y= 2x^2 -5x +4 at the point where x=1 is parallel to the normal to the curve y= ax^2 + bx +10 at the point (-2,2). Calculate the values of a and b.
The answers are a=1, b=6
yes, that is what I get also.
The answer is at the back of the book but I cannot understand how to get to that answer!
To solve this problem, let's start by finding the gradient of the tangent to the curve 2y = 2x^2 - 5x + 4 at the point (1, y).
To find the gradient, we can differentiate the equation of the curve with respect to x:
d/dx (2y) = d/dx (2x^2 - 5x + 4)
This simplifies to:
2(dy/dx) = 4x - 5
Now, let's find the value of dy/dx at x=1 by substituting it into the equation:
2(dy/dx) = 4(1) - 5
2(dy/dx) = 4 - 5
2(dy/dx) = -1
dy/dx = -1/2
Therefore, the gradient of the tangent to the curve at the point (1, y) is -1/2.
Now, let's find the gradient of the normal to the curve y = ax^2 + bx + 10 at the point (-2, 2).
To find the gradient, we can differentiate the equation of the curve with respect to x:
dy/dx = d/dx (ax^2 + bx + 10)
This simplifies to:
dy/dx = 2ax + b
Now, substitute the coordinates of the point (-2, 2) into the equation:
2 = 2a(-2) + b
Simplify the equation:
2 = -4a + b
From the given condition in the problem, the tangent and the normal are parallel, which means their gradients are equal. Therefore, we equate the gradients:
-1/2 = -4a + b (Equation 1)
2 = -4a + b (Equation 2)
Now, we can solve this system of equations to find the values of a and b.
Add Equations 1 and 2:
-1/2 + 2 = -4a + b + -4a + b
Simplify:
3/2 = -8a + 2b
Now, rearrange this equation:
8a - 2b = -3/2 (Equation 3)
Multiply Equation 2 by 2:
4 = -8a + 2b (Equation 4)
Add Equations 3 and 4:
8a - 2b + 4 = -3/2 + 4
Simplify:
8a - 2b + 4 = 5/2
Rearrange the equation:
8a - 2b = 5/2 - 4
8a - 2b = (5 - 8)/2
8a - 2b = -3/2
This equation is the same as Equation 3, so the system of equations is consistent.
Now, we can solve the system of equations:
Equation 3: 8a - 2b = -3/2
Equation 4: 4 = -8a + 2b
Let's solve it using the method of elimination:
Multiply Equation 3 by 2:
16a - 4b = -3
Add the two equations:
(16a - 4b) + (4 = -8a + 2b)
Simplify:
16a - 8a - 4b + 2b = -3 - 4
8a - 2b = -7
Now, divide the equation by 2 to simplify it further:
4a - b = -7/2
Rearrange the equation:
b = 4a + 7/2
Substitute this value of b in Equation 4:
4 = -8a + 2(4a + 7/2)
Simplify:
4 = -8a + 8a + 7
Simplify further:
4 = 7
This equation is not possible. It means that there is no unique solution to the system of equations, which implies that the original problem is inconsistent.
Therefore, no values of a and b satisfy the conditions given in the problem statement.