Graph 4x^2-y^2-16x-6y-9=0

Problem

4*x2−y2−16*x−6*y−9=0

Solution

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

4*x2−16*x−y2−6*y=9

2. Factor out the leading coefficients from the x and y groups to prepare for completing the square.

4*(x2−4*x)−(y2+6*y)=9

3. Complete the square for both the x and y expressions by adding and subtracting the necessary constants inside the parentheses.

4*(x2−4*x+4)−(y2+6*y+9)=9+4*(4)−1*(9)

4. Simplify the equation by writing the quadratic expressions as perfect squares and calculating the sum on the right side.

4*(x−2)2−(y+3)2=16

5. Divide both sides by 16 to put the equation into the standard form of a hyperbola.

((x−2)2)/4−((y+3)2)/16=1

6. Identify the key features of the hyperbola: the center is (2,−3) it opens horizontally because the xterm is positive, the horizontal semi-axis is a=√(,4)=2 and the vertical semi-axis is b=√(,16)=4

Final Answer

((x−2)2)/4−((y+3)2)/16=1

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