Question

Sketch the graph of the given equation. \(x^{2}+4 y^{2}-2 x+16 y+1=0\)

Short answer

The equation represents an ellipse centered at (1, -2) with axes of lengths 4 and 2.

Step-by-step solution

Identify the Type of Conic

The given equation is of the form \[ x^2 + By^2 + Dx + Ey + F = 0 \] which suggests it is a conic section. In this case, the coefficients of the squared terms are different, and the product of the coefficients of \(x^2\) and \(y^2\) is positive, indicating it is an ellipse.

Group and Rearrange Terms

Start by rearranging the terms of the equation: \[ x^2 - 2x + 4y^2 + 16y + 1 = 0 \] Group the \(x\)-terms and \(y\)-terms: \[ (x^2 - 2x) + (4y^2 + 16y) + 1 = 0 \]

Complete the Square for x

Take the terms related to \(x\): \[ x^2 - 2x \] To complete the square, add and subtract \(1\): \[ (x^2 - 2x + 1 - 1) \equiv (x - 1)^2 - 1 \]

Complete the Square for y

Factor out \(4\) from the \(y\) terms:\[ 4(y^2 + 4y) \] Complete the square inside the parentheses by adding and subtracting \(4\): \[ 4((y^2 + 4y + 4 - 4)) \equiv 4((y + 2)^2 - 4) \] Simplify to: \[ 4(y + 2)^2 - 16 \]

Combine Completed Squares

Substitute the completed squares back into the equation: \[ (x - 1)^2 - 1 + 4(y + 2)^2 - 16 + 1 = 0 \] Simplify to: \[ (x - 1)^2 + 4(y + 2)^2 = 16 \]

Write the Equation of an Ellipse

Divide the entire equation by \(16\) to express it in standard form: \[ \frac{(x - 1)^2}{16} + \frac{(y + 2)^2}{4} = 1 \] The equation is now in the form of an ellipse with center \((1, -2)\), semi-major axis length \(4\) along the x-axis, and a semi-minor axis length \(2\) along the y-axis.

Sketch the Graph

Plot the center of the ellipse at \((1, -2)\) on the coordinate plane. From the center, measure 4 units along the x-axis in both directions to locate vertices \((5, -2)\) and \((-3, -2)\). Measure 2 units along the y-axis in both directions to locate co-vertices \((1, 0)\) and \((1, -4)\). Sketch an ellipse conforming to these points.

Key concepts

Conic Sections

Conic sections are fascinating curves that come from slicing a cone with a plane. They include shapes like circles, ellipses, parabolas, and hyperbolas. You can find these curves in many areas of mathematics and the natural world. For example, planetary orbits are ellipses! In equations, conic sections have the general form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] When there's only one squared term or the coefficients differ, it's usually a parabola, ellipse, or hyperbola. The given equation: \[ x^2 + 4y^2 - 2x + 16y + 1 = 0 \] initially looks complex, but by studying the coefficients, we recognize it as an ellipse because the squared terms have different coefficients, and their product is positive. This step helps us determine the type of curve just by analyzing the equation.

Completing the Square

Completing the square is a valuable algebraic technique. It transforms quadratic equations, making them easier to work with. This method rewrites a quadratic expression, like \[ x^2 - 2x \] into a perfect square trinomial. By carefully adding and subtracting terms, we turn it into \[ (x - 1)^2 - 1 \]. This converts unwieldy equations into a more manageable form. Similarly, for the equation's y-terms \[ 4(y^2 + 4y) \], we first factor out the 4 and then complete the square inside the parentheses. This becomes \[ 4((y + 2)^2 - 4) \]. Completing the square is crucial as it helps express the ellipse's equation in a form that's easier to interpret.

Standard Form of Ellipse

The standard form of an ellipse's equation is incredibly useful for graphing. Once we have completed the square, we can write the equation like this:\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \], where

• \((h, k)\) is the center of the ellipse,
• \(a\) is the semi-major axis length,
• \(b\) is the semi-minor axis length.

In our exercise, after transforming and simplifying, we get the ellipse:\[ \frac{(x - 1)^2}{16} + \frac{(y + 2)^2}{4} = 1 \]. Here, the center is \((1, -2)\), stretching 4 units along the x-axis, and 2 units along the y-axis. This form precisely identifies key features, making graphing straightforward.

Graphing Techniques

Graphing an ellipse is quite simple once it is in standard form. Identify the center, \( (1, -2) \), and draw it on the coordinate plane. From this point, mark 4 units left and right on the x-axis to find the vertices \((5, -2)\) and \((-3, -2)\).Do the same along the y-axis, moving 2 units up and down to locate the co-vertices at \((1, 0)\) and \((1, -4)\).Next, sketch the ellipse. Practice makes perfect! The more you graph, the more natural it will become. Understanding these steps builds a solid foundation for mastering more complex curves later.