Question

The center of the equation below is in the form (h,k), find h+k.

Answer 1

The function given in the question is

`9x^2+y^2+18x-12y-180=0`

Rearrange the function above connecting similar terms

`\begin{gathered} 9x^2+y^2+18x-12y-180=0 \\ 9x^2+18x+y^2-12y=180 \end{gathered}`

Factorize the equation in brackets

`\begin{gathered} 9x^2+18x+y^2-12y=180 \\ 9(x^2+2x)+1(y^2-12y)=180 \\ \end{gathered}`

Solve the brackets using completing the square method but multiplying the coefficient of x and y by 1/2 and then square

`\begin{gathered} 9(x^2+2x)+1(y^2-12y)=180 \\ 9(x^2+2x+(2*(1)/(2))^2)+1(y^2-12y+(-12*(1)/(2))^2=180+(2*(1)/(2))^2+(-12*(1)/(2))^2 \end{gathered}`

By simplifying the equation above, we will have

`\begin{gathered} 9(x^2+2x+(2*(1)/(2))^2)+1(y^2-12y+(-12*(1)/(2))^2=180+9(2*(1)/(2))^2+1(-12*(1)/(2))^2 \\ 9(x^2+2x+1)+1(y^2-12y+36)=180+9+36 \\ 9(x+1)^2+1(y-6)^2=225 \end{gathered}`

Divide all through by 225

`\begin{gathered} 9(x+1)^2+1(y-6)^2=225 \\ (9(x+1)^2)/(225)+(1(y-6)^2)/(225)=(225)/(225) \\ ((x+1)^2)/(25)+((y-6)^2)/(225)=1 \end{gathered}`

The general formula for the equation of an ellipse is

`((x-h)^2)/(a^2)+((y-k)^2)/(b^2)=1`

Where the center of the ellipse is

`(h,k)`

By comparing coefficients, we will have that

`\begin{gathered} x+1=x-h,y-6=y-k \\ x-x+h=-1,y-y+k=6 \\ h=-1,k=6 \end{gathered}`

The centre of the ellipse is

`\begin{gathered} (h,k)=(-1,6) \\ \text{hence,} \\ h+k=-1+6 \\ h+k=5 \end{gathered}`

Therefore,

The value of h+k = 5