# Find the Value of k?

Find k such that the given line is tangent to the graph of the function.

f(x) = k - x^2

Given Line: y = 4x - 9

Relevance

f(x) = k - x² ← this is the curve

y = 4x - 9 ← this is the line

If the line is tangent to the curve, it means that there is only one common point between the line and the curve.

f(x) = y → gives us only one point, so only one solution to the equation

4x - 9 = k - x²

x² + 4x - 9 - k = 0

x² + 4x - (9 + k) = 0

Polynomial like: ax² + bx + c, where:

a = 1

b = 4

c = - (9 + k)

Δ = b² - 4ac (discriminant)

Δ = 4² - 4.[1 * - (9 + k)]

Δ = 16 + 4.(9 + k)

Δ = 16 + 36 + 4k

Δ = 4k + 52 → only one solution → Δ = 0

4k + 52 = 0

4k = - 52

→ k = - 13

The solution is: x = - b/2a

x = - 4/2

→ x = - 2

• f(x) = k - x^2. L is line y = 4x - 9. What is k if L is tangent to f(x)? If we solve the eqn y=f(x) we get k-x^2 -4x+9

= 0, ie., x^2 + 4x - (k+9) = 0 whose discriminant = 4^2 + 4(k+9) = 4[k+13] = 0 for equal roots which occur when

L is tangent to f(x).. Thus k = - 13. End of story.

• If the line is a tangent to the curve, they meet only once, and the equation

k - x^2 = 4*x - 9

has only one solution.

x^2 + 4*x - (9 + k) = 0

For only one solution,

(coefficient of x)squared = 4*coefficient of x*constant coefficient.

16 = -4*(9 + k)

16 = -36 - 4*k

k = -13 <<<

• I do not know exactly what did you mean

• PhotonX
Lv 7
4 years agoReport

She's a point w h o r e, Jysero. Report junk answers--that's what the pop up flag on the right side of the question is for.