For what values of a and b is the line 2x+y=b tangent to the parabola y=ax^2 when x=2?

Question

I know how to do this when just given a parabola and points but I'm confused how to work it backwards. I need help going through the process not just the answer.

Answer 1

The slope of the parabola at x = 2 is equal to: 2 a x = 4 a
The slope of the line is -2, so if it is the tangent we must have 4 a = -2, or a = -1/2.

Moreover, since y = ax^2 = -1/2 * 2^2 = -2, the line goes through the point (2, -2). Therefore,

2x + y = b
2*2 + (-2) = b
b = 4-2 = 2.

Answer 2

Because of the way the problem is setup, the line already has a slope -- -2(rearanging to y = -2 + b). All we can modify on the line is its y-intercept. On the parabola, however, we can modify its steepness by changing a. So, the plan is change b to allow the line to intersect the parabola at x=2, and change a to adjust the slope of the parabola at x=2 to be equalto the slope of the line.

Lets do the second part first. Differentiating the function of the parabola,we get:
y'=2ax y' represents the slope of the parabola at some point x.

So, since we need the slope to be -2 at x=2:
-2 = 2a2
-2 = 4a
a = -1/2

Now, lets find the y coordinate on the parabola when x=2:
y = (-1/2)*(2)^2
y = -4/2 = -2

So, now all we have to do is change b so that the line's y coordinate is 2 when x = 2
y = -2x + b
-2 = -2*2 + b
b = 2

Answer 3

I could be wrong, but I think this is the way. To find the slope of the tangent, take the derivative of the equation for the parabola, dy/dx = 2ax . Then the tangent has to intersect the parabola at the same point so putting the linear equation in the standard term, y=mx+b where m is the slope and b is the y intercept, y= -2x + b. Solving for the general slopes, 2ax= -2x (and x is not 0) so a= -1. Now at x=2, a=-1, the parabolic equation gives y = -4. Putting x and y in the linear equation, 4 + -4 = b = 0. I hope I'm right. Graphing it out would tell for sure.

Answer 4

Find slope of parabola at x=2.
For that differentiate both sides of eqn. with respect to x.
so, slope is dy/dx=2ax
The slope of the parabola at x = 2 is equal to: 2 a x = 4 a
The slope of the line is -2, so if it is the tangent we must have 4 a = -2, or a = -1/2.

Moreover, since y = ax^2 = -1/2 * 2^2 = -2, the line goes through the point (2, -2). Therefore,

2x + y = b
2*2 + (-2) = b
b = 4-2 = 2.

Answer 5

I can see by the type of questions you ask that you are in a calculus course. I am assuming you understand that the derivative of a function will give its slope. Let's start from there...

Put the line 2x+y=b into slope-intercept form: y=-2x+b. Thus we see that this line has slope m=-2. Now, where does our parabola have a slope of -2? We'll take its derivative.

y=ax^2

y'=2ax We want this slope to be -2, like our line, when x=2

2ax=-2 now substitute 2 for x
2a(2)=-2 now solve for a
4a=-2
a=-1/2

The equation of the parabola is therefore y=(-1/2)x^2

Finding the ordered pair on the parabola for when x=2 you get (2, -2)
This is the tangent point that is shared by both lines so (2,-2) must also work with the straight line. We can substitute this data into the line's equation and solve for b.

2x+y=b
2(2)-2=b
4-2=b
b=2

a=-1/2
b=2

The line is 2x+y=2
The parabola is y=(-1/2)x^2

Answer 6

slope of the given line is -2

Slope at any point on the parabola is dy/dx= 2ax

Slope at x=2 is 4a

therefore 4a= -2, a=-1/2

Also the line will be a tangent if it passes through (2, -2)

i.e. 4-2=b,
b=2

Answer 7

parabola
-----------
p(x) = a*x^2
p'(x) = 2*a*x
p'(2) = 4*a

line
----
l(x) = -2*x + b
l'(x) = -2
l(2) = b - 4

=====
Need slope of line to equal slope of parabola at x=2, so,
4a = p'(2) = l'(2) = -2 or,
a = -1/2.
(so p(2) = -2, needed in next step).

And need line to intersect parabola at x=2 so need,
-2 = p(2) = l(2) = b - 4 or,
b = 2

Answer 8

y = ax^2, dy/dx = 2ax ,when x =2, dy/dx = 4a.

by substituting the value of 'x' into y =ax^2, we get4a.

so, the point of contact is (2, 4a)

y - y1 = m ( x - x1) is equation .

we get y1 is 4a and x1 is 2.

y- 4a = 4a ( x - 2 )

y - 4a = 4ax - 8a

y + 4a = 4ax, according to equation given, y + 2x = b

y-4ax = -4a, y + 2x = b

so, -4ax = 2x

a = -1/2

when a =-1/2, b = -4a

b = -4 x -1/2

b = 2

so, 'a' is -1/2 and 'b' is 2.

Answer 9

y = ax² y ´= 2ax 2x + y = b --> y(tg) = - 2x + b --> m(tg) = - 2 y ´(2) = - 2 --> 4a = - 2 --> a = - a million/2 at x = 2, ax² = - 2x + b --> (- a million/2) 2² = - 4 + b --> b = 2 Alejandra