Question

identify the type of conic section given by the following equation:

\\16x^2 - 25y^2 - 192x + 150y - 49 = 0\\

• parabola
• ellipse
• hyperbola
• degenerate conic

Explanation:

Step1: Group terms

Group the \(x\) and \(y\) terms together.

$$(16x^2 - 192x) - (25y^2 - 150y) = 49$$

Step2: Factor coefficients

Factor out the leading coefficients from each group.

$$16(x^2 - 12x) - 25(y^2 - 6y) = 49$$

Step3: Complete squares

Add the necessary constants to complete the squares.

$$16(x^2 - 12x + 36) - 25(y^2 - 6y + 9) = 49 + 16(36) - 25(9)$$

Step4: Simplify equation

Simplify both sides of the equation.

$$16(x - 6)^2 - 25(y - 3)^2 = 400$$

Step5: Standard form

Divide by 400 to get the standard form.

$$\frac{(x - 6)^2}{25} - \frac{(y - 3)^2}{16} = 1$$

Answer:

Hyperbola