Question

the equation of the ellipse is given:

\\\frac{(x - 4)^2}{144} + \frac{(y + 2)^2}{196} = 1\\

what is the major radius? units

what is the minor radius? units

Explanation:

🆕 New Concept Discovered: Equations and Properties of Ellipses
The larger denominator determines the major axis.

Step 1: Identify the standard form of the ellipse

An ellipse centered at \( (h, k) \) has one of two standard forms depending on whether it is horizontal or vertical:

$$ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \quad \text{or} \quad \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 $$

where \( a \) represents the semi-major axis (major radius) and \( b \) represents the semi-minor axis (minor radius), with \( a > b \).

Comparing this to our given equation:

$$ \frac{(x-4)^2}{144} + \frac{(y+2)^2}{196} = 1 $$

We identify the denominators:

• Under the \( x \)-term: \( 144 \)
• Under the \( y \)-term: \( 196 \)

Step 2: Determine the major radius

Since \( 196 > 144 \), the larger value corresponds to the square of the major radius, \( a^2 \):

$$ a^2 = 196 $$

To find the major radius \( a \), we take the square root:

$$ a = \sqrt{196} = 14 $$

Step 3: Determine the minor radius

The smaller value corresponds to the square of the minor radius, \( b^2 \):

$$ b^2 = 144 $$

To find the minor radius \( b \), we take the square root:

$$ b = \sqrt{144} = 12 $$

Answer:

• What is the major radius? 14 units
• What is the minor radius? 12 units