Problem 52 - Algebra 2

Question

What is the standard form of an equation of a hyperbola? Explain how to rewrite \(25 x^{2}-49 y^{2}=1225\) in standard form.

Step-by-Step Solution

Answer

The standard form of the given equation is \(x^2/49 - y^2/25 = 1\), with \(a = 7\) and \(b = 5\).

Step 1: Identify the form of the hyperbola

The given equation is \(25x^2 - 49y^2 = 1225\). Notice that the equation includes a difference of squares, so we are dealing with a hyperbola. Since the term with \(x^2\) has a positive coefficient and is first, this is a hyperbola that opens horizontally.

Step 2: Rearrange into standard form

To put this into standard form, we need it in the form \((x^2/a^2) - (y^2/b^2) = 1\). This means we need to divide each term by \(1225\) (the constant on the right-hand side) to make the right-hand side \(1\). This gives us: \((25x^2/1225) - (49y^2/1225) = 1\) or equivalently \(x^2/49 - y^2/25 = 1\)

Step 3: Identify the values of a and b

From the rearranged equation, we can see that \(a^2 = 49\) and \(b^2 = 25\). Therefore, \(a = 7\) and \(b = 5\). These values tell us more about the hyperbola such as the steepness of the opening. The larger \(a\) and \(b\) are, the 'flatter' the hyperbola.

Key Concepts

Standard Form

In mathematics, understanding the standard form of an equation is crucial when dealing with conic sections like hyperbolas. The standard form for the equation of a hyperbola is expressed as \((x^2/a^2) - (y^2/b^2) = 1\) or \((y^2/a^2) - (x^2/b^2) = 1\), depending on the orientation of the hyperbola.
For a hyperbola that opens horizontally, we use the first form. This involves rearranging the equation so that similar terms like squares of variables and coefficients are easily comparable.
Each term is divided by a constant to ensure the right side of the equation equals one, forming this neat, clean expression. The form helps simplify calculations and analyses, making it easier to graph and understand the hyperbola's properties.

Conic Sections

Conic sections arise from slicing a cone with a plane at different angles. These intersections create various curves: circles, ellipses, parabolas, and hyperbolas, each with distinct properties.
Hyperbolas are among these intriguing shapes, distinguishable by their two separate branches. They occur when the cone is cut in such a way that the plane intersects both halves of the double cone.
This specific cut leads to equations that involve subtraction, indicating a 'difference', a characteristic feature of hyperbolas. Recognizing the type of conic section is vital in solving and interpreting related mathematical problems.

Difference of Squares

The concept of the difference of squares is foundational in understanding hyperbolas. A difference of squares refers to an expression of the form \(A^2 - B^2\), which factors into \((A + B)(A - B)\).
In the equation of a hyperbola, this characteristic is directly translated into the subtracted squares \(x^2 - y^2\) or \(y^2 - x^2\).
Understanding this difference helps distinguish hyperbolas from other conic sections. Its recognition allows us to restructure equations correctly into their standard forms, highlighting the symmetry and orientation of the hyperbola's branches.

Equation of a Hyperbola

Deriving the equation of a hyperbola involves identifying key components that shape its graph. The general form you'll often start with looks something like \(Ax^2 + By^2 + C = 0\).
Transforming this into the standard form involves isolating terms, utilizing the difference of squares, and dividing by constants appropriately.

• The term with the positive square indicates the direction the hyperbola opens.
• The values \(a\) and \(b\) define the 'stretch' or 'compression' of the hyperbola in respective directions.
• The larger these values, the more spread out the hyperbola appears.

Completing these steps lets you graph the hyperbola accurately, understanding both its orientation and eccentricity.