Subtraction: fill in the missing digits

Learning Objectives Analyze the standard form of a hyperbola to identify how subtraction dictates its orientation and key features. Apply the technique of completing the square to find missing coefficients in the general form of a hyperbola's equation. Use the relationship c² = a² + b² to calculate missing parameters (a², b², or c²) of a hyperbola through subtraction. Determine the missing coordinates of a hyperbola's center, vertices, or foci given an incomplete equation. Reconstruct the full equation of a hyperbola by using subtraction to deduce missing values from given geometric properties. Solve for unknown variables in problems involving the geometric definition of a hyperbola, where the difference of distances is constant. An asteroid is streaking through th...

TermDefinitionExample Standard Equation (Subtraction Form)The defining equation of a hyperbola where the squared terms for the variables are subtracted from one another, dictating the shape and orientation of the curve.In the equation \frac{(x-2)^2}{9} - \frac{(y-1)^2}{16} = 1, the subtraction of the y-term indicates a horizontal hyperbola. Center (h, k)The point of symmetry for a hyperbola, determined by the values being subtracted from x and y in the standard form.For the hyperbola \frac{(x-5)^2}{4} - \frac{(y+3)^2}{9} = 1, the center is (5, -3), because y+3 is equivalent to y - (-3). Transverse AxisThe axis that passes through the center, foci, and vertices of a hyperbola. The variable in the positive (minuend) term of the standard equation determines its orientation.In \frac{x^2}{25}...

Standard Equation of a Horizontal Hyperbola \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 Use this formula when the transverse axis is horizontal. The key feature is the subtraction of the y-term from the x-term. Standard Equation of a Vertical Hyperbola \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 Use this formula when the transverse axis is vertical. The key feature is the subtraction of the x-term from the y-term. Focal Length Formula c^2 = a^2 + b^2 This formula connects the distance from the center to a vertex (a), the conjugate axis parameter (b), and the distance from the center to a focus (c). It is essential for finding a missing parameter when two are known, often by rearranging it as a subtraction problem (e.g., b² = c² - a²).