Question

How do you change `x^2 + 4y^2 - 4x + 8y - 60 = 0` into standard form?

Answer 1

`((x-2)^2)/((2sqrt(17))^2)+((y+1)^2)/((sqrt(17)/2)^2)=1`

Explanation:

The given equation is already in standard polynomial form, so I assume that what is required is to express this in standard elliptical form.
`color(white)("XXX")((x-h)^2)/(a^2) +((y-k)^2)/(b^2)=1`

Given:
`color(white)("XXX")x^2+4y^2-4x+8y-60=0`

Re-arranging the terms to group the `x` terms, the `y` terms, and shift the constant to the right side:
`color(white)("XXX")x^2-4x+4y^2+8y=60`

Extract the common factor from the `y` terms and complete the squares
`color(white)("XXX")x^2-4x+4+4(y^2+2y+1)=60+4+4`

Re-writing as squared binomials and simplifying
`color(white)("XXX")(x-2)^2+4(y+1)^2=68`

Divide through by 68and take roots of denominators
`color(white)("XXX")((x-2)^2)/((2sqrt(17))^2)+((y+1)^2)/((sqrt(17)/2)^2)=1`