I'm unsure of how to do this. Walk me through it?
Identify the conic section represented by each equation.
x^2 - 4y^2 + 4x + 8y - 16 = 0
x^2 - 4x - y + 7 = 0
x^2 + y^2 - 4x + 2y + 1 = 0
What is the distance between P(3, -2) and Q(-5, -2)?
To identify the conic section represented by each equation, we need to examine the equations and determine their general form.
The general form of the equation for a conic section is often presented as:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
By comparing the given equations to this general form, we can identify the conic section. Let's go through each equation step by step.
Equation 1: x^2 - 4y^2 + 4x + 8y - 16 = 0
To identify the conic section using this equation, we can rewrite it in the general form:
x^2 + 4x - 4y^2 + 8y = 16
Rearranging the terms:
x^2 + 4x + 4 - 4y^2 + 8y + 4 = 16 + 4
Factoring:
(x + 2)^2 - 4(y - 1)^2 = 20
By comparing this equation to standard conic section equations, we see that it matches the equation of a hyperbola. Therefore, Equation 1 represents a hyperbola.
Equation 2: x^2 - 4x - y + 7 = 0
Rearranging the terms:
x^2 - 4x = y - 7
Comparing it to the general form, we don't have any xy term. Hence, the equation does not contain a conic section.
Equation 3: x^2 + y^2 - 4x + 2y + 1 = 0
Rearranging the terms:
x^2 - 4x + y^2 + 2y = -1
Completing the squares for both x and y terms:
(x^2 - 4x + 4) + (y^2 + 2y + 1) = -1 + 4 + 1
(x - 2)^2 + (y + 1)^2 = 4
Comparing it to the general form, we can see that it matches the equation of a circle. Therefore, Equation 3 represents a circle.
To summarize:
- Equation 1 represents a hyperbola.
- Equation 2 does not represent a conic section.
- Equation 3 represents a circle.