Find the Domain 16x^2+25y^2+32x^2-100y-284=0

Problem

16*x2+25*y2+32*x−100*y−284=0

Solution

1. Group the x and y terms together to prepare for completing the square.

16*x2+32*x+25*y2−100*y=284

2. Factor out the leading coefficients from the x and y groups.

16*(x2+2*x)+25*(y2−4*y)=284

3. Complete the square for both variables by adding (2/2)2=1 to the x group and ((−4)/2)2=4 to the y group.

16*(x2+2*x+1)+25*(y2−4*y+4)=284+16*(1)+25*(4)

4. Simplify the right side and write the left side as squared binomials.

16*(x+1)2+25*(y−2)2=400

5. Divide the entire equation by 400 to put the ellipse equation in standard form.

((x+1)2)/25+((y−2)2)/16=1

6. Identify the center (h,k) and the horizontal semi-axis a from the standard form ((x−h)2)/(a2)+((y−k)2)/(b2)=1

h=−1

a2=25⇒a=5

7. Determine the domain by calculating the interval [h−a,h+a]

−1−5≤x≤−1+5

−6≤x≤4

Final Answer

Domain:[−6,4]

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