Question

4. Use the hyperbola equationand y-values in the table.(2 - 21) (y - yı)a262= 1 to find the y-values to the nearest integer from the gis(2-2)²9( 1)4y6210100 - 40-6-202

Answer 1

Isolate y from the equation:

`((x-2)^2)/(9)-((y-1)^2)/(4)=1`
`\Rightarrow-((y-1)^2)/(4)=1-((x-2)^2)/(9)`
`\begin{gathered} \Rightarrow(y-1)^2=(1-((x-2)^2)/(9))(-4) \\ =(4)/(9)(x-2)^2-4 \end{gathered}`
`\begin{gathered} \Rightarrow y-1=\pm\sqrt[]{(4)/(9)(x-2)^2-4} \\ \Rightarrow y=1\pm\sqrt[]{(4)/(9)(x-2)^2-4} \end{gathered}`

Substitute x=10 to find the two possible values for y:

`\begin{gathered} y=1\pm\sqrt[]{(4)/(9)(10-2)^2-4} \\ =1\pm\sqrt[]{(4)/(9)(8)^2-4} \\ =1\pm\sqrt[]{(4)/(9)(64)^{}-4} \\ =1\pm\sqrt[]{(256)/(9)-4} \\ =1\pm\sqrt[]{(256-36)/(9)} \\ =1\pm\sqrt[]{(220)/(9)} \\ =1\pm\frac{2\sqrt[]{55}}{3} \end{gathered}`

The aproximate values of y are:

`\begin{gathered} y_1\approx5.944\ldots \\ y_2\approx-3.944\ldots \end{gathered}`

To the nearest whole number, we can see that the second value of y is approximately -4.