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

Answer:

k = 9

length of chord = 2/3

Step-by-step explanation:

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

Part 1

If the curve passes through point $(\frac23 ,1)$, this means that when $x=\dfrac23$, $y = 1$

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

$\implies 1=k \left(\dfrac23-\dfrac13\right)^2$

$\implies 1=k \left(\dfrac13 \right)^2$

Apply the exponent rule $\left(\dfrac{a}{b} \right)^c=\dfrac{a^c}{b^c}$ :

$\implies 1=k \left(\dfrac{1^2}{3^2} \right)$

$\implies 1=\dfrac{1}{9}k$

$\implies k=9$

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:

$\implies 1=9 \left(x-\dfrac13 \right)^2$

$\implies \dfrac19 = \left(x-\dfrac13 \right)^2$

$\implies \sqrt{\dfrac19} = x-\dfrac13$

$\implies \pm \dfrac13 = x-\dfrac13$

$\implies x=\dfrac23, x=0$

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:  

$\implies \dfrac23-0=\dfrac23$

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