Question

Suppose that y = k * (x - 1/3) ^ 2 is a parabola in the xy -plane that passes through the point (2/3, 1) :Find k and the length of the horizontal chord of the parabola that has one end at (2/3, 1) .

Answer 1

k = 9

length of chord = 2/3

Explanation:

Equation of parabola:
`y=k (x-\frac13)^2`

Part 1

If the curve passes through point
`(\frac23 ,1)`, this means that when
`x=\frac23`,
`y = 1`

Substitute these values into the equation and solve for
`k`:

`\implies 1=k \left(\frac23-\frac13\right)^2`

`\implies 1=k \left(\frac13 \right)^2`

Apply the exponent rule
`\left((a)/(b) \right)^c=(a^c)/(b^c)` :

`\implies 1=k \left((1^2)/(3^2) \right)`

`\implies 1=(1)/(9)k`

`\implies k=9`

Part 2

• The chord of a parabola is a line segment whose endpoints are points on the parabola.

We are told that one end of the chord is at
`(\frac23 ,1)` and that the chord is horizontal. Therefore, the y-coordinate of the other end of the chord will also be 1. Substitute y = 1 into the equation for the parabola and solve for x:

`\implies 1=9 \left(x-\frac13 \right)^2`

`\implies \frac19 = \left(x-\frac13 \right)^2`

`\implies √(\frac19) = x-\frac13`

`\implies \pm \frac13 = x-\frac13`

`\implies x=\frac23, x=0`

Therefore, the endpoints of the horizontal chord are: (0, 1) and (2/3, 1)

To calculate the length of the chord, find the difference between the x-coordinates:

`\implies \frac23-0=\frac23`

**Please see attached diagram for drawn graph. Chord is in red**