For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 5?
for B the answer would be 5, the point (5,-5) when plugged into the the 2x+y=b equation gives you 2(5)+(-5)= b
b=5
Well, to find the values of a and b, let's take a humorous approach!
Let's start by assuming that the line and the parabola meet at the point (5, y). Since the line is tangent to the parabola, this means that they share the same y-coordinate.
Now, let's substitute the values into the equations.
For the line: 2x + y = b
If we plug in x = 5, we get 2(5) + y = b, which simplifies to y = b - 10.
For the parabola: y = ax^2
Since the line and the parabola share the same y-coordinate, we can equate the two equations:
b - 10 = a(5^2)
b - 10 = 25a
Now, we can solve for a and b!
So, the values of a and b for which the line 2x + y = b is tangent to the parabola y = ax^2 when x = 5 are when a = (b - 10)/25.
Now, go out there and have some tangent-al laughs with your equations!
To determine the values of a and b for which the line 2x + y = b is tangent to the parabola y = ax² when x = 5, we need to find the point of tangency between the line and the parabola.
Step 1: Substitute x = 5 into the equation of the parabola to find the corresponding y-coordinate.
Substitute x = 5 into y = ax²:
y = a(5)² = 25a
Step 2: Determine the derivative of the parabola.
To find the slope of the tangent line, we need to take the derivative of y = ax².
dy/dx = 2ax
Step 3: Substitute x = 5 into the derivative equation to find the slope at x = 5.
dy/dx = 2a(5) = 10a
Step 4: Set the slope of the tangent line equal to the slope obtained from the derivative.
The slope of the tangent line is -2 (since the coefficient of x in the line equation is 2), so we equate it to 10a:
10a = -2
Step 5: Solve for a.
Divide both sides of the equation by 10:
a = -2/10
a = -1/5
Step 6: Substitute the values of x = 5 and a = -1/5 into the equation of the parabola to find the y-coordinate of the point of tangency.
y = (-1/5)(5)² = (-1/5)(25) = -5
Step 7: Substitute the values of x = 5 and y = -5 into the equation of the line to find the value of b.
2(5) + (-5) = b
10 - 5 = b
b = 5
Therefore, the line 2x + y = 5 is tangent to the parabola y = (-1/5)x² when x = 5. So, the values of a and b are a = -1/5 and b = 5.
the slope of 2x+y = b is -2
the slope of y = ax^2 is
dy/dx = 2ax
so when x = 5
2a(5) = -2
a = -2/10 = -1/5
so the parabola is y = (-1/5)x^2
when x=5, y = -5
but that point (5,-5) must also be on the tangent
2(5)+5 = b
b = 15
so a= -1/5 and b=15