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Chapter 10: Problem 32

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

Short Answer

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

Step by step solution

01

Write out the given equation

The conic section equation given is \(10 x^{2}+24 x y+17 y^{2}-9=0\)
02

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.
03

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 \( (24)^2 - 4*10*17 = 576 - 680 = -104 \)
04

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.

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