Problem 32 - Algebra and Trigonometry

Question

Identify each equation without applying a rotation of axes. $$10 x^{2}+24 x y+17 y^{2}-9=0$$

Step-by-Step Solution

Answer

The given equation is of an ellipse as the discriminant is less than 0.

Step 1: Write out the given equation

The conic section equation given is \(10 x^{2}+24 x y+17 y^{2}-9=0\)

Step 2: Identify the coefficients of the given equation

The equation contains the expression in form \(Ax^2 + Bxy + Cy^2 = D\). Identify the following coefficients: A = 10, B = 24, C = 17 and D = 9.

Step 3: Calculate the Discriminant

The discriminant of an equation of this type can be found using the formula \(B^2 - 4AC\). Substituting in the identified coefficients produces \( (24)^2 - 4*10*17 = 576 - 680 = -104 \)

Step 4: Determine the type of conic section based on the Discriminant

If the discriminant is less than 0, the equation is of an ellipse. Since the discriminant calculated is -104, which is less than 0, the equation represents an ellipse.