Find the Domain -16x^2+y^2-32x-14y+17=0

Problem

−16*x2+y2−32*x−14*y+17=0

Solution

1. Group the terms containing x and the terms containing y together, and move the constant to the other side of the equation.

−16*x2−32*x+y2−14*y=−17

2. Factor out the leading coefficient from the x terms to prepare for completing the square.

−16*(x2+2*x)+(y2−14*y)=−17

3. Complete the square for both the x and y expressions by adding and subtracting the square of half the linear coefficients.

−16*(x2+2*x+1−1)+(y2−14*y+49−49)=−17

−16*(x+1)2+16+(y−7)2−49=−17

4. Simplify the equation by combining the constants and moving them to the right side.

−16*(x+1)2+(y−7)2−33=−17

−16*(x+1)2+(y−7)2=16

5. Divide the entire equation by 16 to put the hyperbola equation into standard form.

((y−7)2)/16−((x+1)2)/1=1

6. Identify the type of conic section and its orientation. This is a vertical hyperbola opening upward and downward.

(y−7)2=16*(1+(x+1)2)

7. Determine the domain by observing the possible values for x In a vertical hyperbola of the form ((y−k)2)/(a2)−((x−h)2)/(b2)=1 the variable x can take any real value because the squared term (x+1)2 is added to 1 when solving for y which will always result in a non-negative value under a square root.

y=7±4√(,1+(x+1)2)

Final Answer

Domain:(−∞,∞)

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