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Chapter 16: Problem 20

The equation \(x^{2}+4 x y+4 y^{2}+2 x=10\) represents which conic section? a) circle b) ellipse c) parabola d) hyperbola

Short Answer

Expert verified
Answer: The given equation represents a parabola.

Step by step solution

01

Write the equation in standard form

Rewrite the given equation in the standard form for conic sections: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, the given equation is \(x^{2}+4 x y+4 y^{2}+2 x=10\). We can rewrite it as: \(x^{2}+4xy+4y^{2}+2x-10=0\) , where A=1, B=4, C=4, D=2, and E=0.
02

Calculate the values needed to determine the conic section type.

Calculate the values of \(A+C\) and \(B^{2} - 4AC\), which will help us determine the type of conic section. Let's calculate A+C: \(A+C=1+4=5\) Now, let's calculate the value of \(B^{2}-4AC\): \(B^{2}-4AC = 4^2 - 4(1)(4) = 16 - 16 = 0\)
03

Determine the type of conic section based on the calculated values.

Using the calculated values of \(A+C\) and \(B^{2}-4AC\), we can determine the type of conic section based on the following conditions: 1. If \(B^{2}-4AC<0\), the equation represents an ellipse (if \(A=C\) and \(B=0\), it is a circle) 2. If \(B^{2}-4AC=0\), the equation represents a parabola 3. If \(B^{2}-4AC>0\), the equation represents a hyperbola Since \(B^{2}-4AC = 0\), the given equation represents a parabola. The correct answer is c) parabola.

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