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Question

3. for what value(s) of k is the line 2x = 3y + k a tangent to the parabola y = x squared - 3x + 4?

I know that i have got to equate the two solutions as they both are equal to y. After solving that, i got a quadratic equation. Using the equation I substituted it into the discriminant (b squared - 4ac = 0). I got the the value of k as -23. But according to the text book im using it says, the correct answer to this question is -23/12. I dont knw where that 12 comes from. Could you please show me how with clear and detailed steps??

Thanks in advance!!

Answer 1

First solve the equation in terms of y:
2x = 3y + k
3y = 2x - k
y = 2/3x - k/3

Now equate to the second equation:
y = x² - 3x + 4
y = 2/3x - k/3

Set everything to one side:
x² - 3x + 4 - 2/3x + k/3 = 0

Group terms:
x² - 11/3x + (4 + k/3) = 0

Now it's probably easier to multiply everything by 3 to get rid of the denominators:
3x² - 11x + (12 + k) = 0

The coefficients are:
a = 3
b = -11
c = (12 + k)

Discriminant = 0
b² - 4ac = 0
(-11)² - 4(3)(12 + k) = 0
121 - 12(12 + k) = 0
121 - 144 - 12k = 0 <--- here's where you probably had k instead of 12k.
-23 - 12k = 0
12k = -23
k = -23/12

Answer 2

Here's another way to do this without quadratic equations.
1). Solve the equation of the line for y:
y = 2/3 x -k/3. So the slope of the line is 2/3.

2). The slope of the tangent line to the parabola at
any point is the derivative of the function or 2x - 3.

3). So 2x -3 = 2/3 and x = 11/6.

4). To find y, plug this value into the equation of the parabola
to get y = 121/36 - 11/2 + 4 = 67/36.

5). Plug both these values into the equation of the
line to find k: k = 11/3 -67/12 = -23/12.

Hope that helps!

Answer 3

Ok let's try this:

Write the equation for the line as
y = 2x/3 - k/3

Set it equal to the parabola:
2x/3 - k/3 = x^2 - 3x + 4

Multiply through by 3:
2x - k = 3x^2 - 9x + 12

Rearrange:
3x^2 - 11x + 12+k = 0

For a tangent, we must have

11^2 - 4*3*(12+k) = 0

for the discriminant to be 0 and have a double root.

We have 121 - 12(12+k) = 0, or
121 - 144 - 12k = 0, or
-23 - 12k = 0,
or -12k = 23,
so k = -23/12.

Answer 4

LOL.

Its a calculation error, my friend.

After substituting the equations you should get something like this,

3x^2 -11x +12 + k

Thus, a = 3, b = -11, c = 12 + k.
Using the discriminant,

b^2 - 4ac = 0
(-11)^2 - 4(3)(12 + k) = 0
121 - 144 -12k = 0
- 23 - 12k = 0
12k = -23
k = -23/12

There you have it. You probably forgot to divide the value of -23 by 12.

Answer 5

Not exactly.
To have a tangent, slopes must be equal.
2x=3y+k
3y=2x-k
y=(2/3)x-k/3 so the slope is 2/3
y=x^2+3x+4
y'=2x+3
y'=2/3=2x+3
2x=(2/3)-3=(2-9)/3
x=-7/6
y=(-7/6)^2+3(-7/6)+4=49/36-21/6+4=(49-126+144)/36=67/36

to find k

2(-7/6)=3(67/36)+k multiply everything by 12
-28=67+12k
12k=-28-67=-95
k=-95/12

I don't know how they got -23/12 im the book
I have known back of the book answers to be wrong.

Answer 6

3/2 = 2X -3
==>X=9/4
PUT X in parabola and get y &the put (x y) in line & get k