Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
Certainly! Let's solve the given equation step-by-step.
The given equation is:
[tex]\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \][/tex]
First, we simplify the denominators:
[tex]\[ 5^2 = 25 \quad \text{and} \quad 3^2 = 9 \][/tex]
Inserting these into the equation, we get:
[tex]\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \][/tex]
Next, we rewrite this equation in a more standard form:
[tex]\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \][/tex]
Now, let us transform this equation to make it look like an equation used in identifying conic sections (like ellipses). Notice that an equation of this form:
[tex]\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \][/tex]
is the standard form of an ellipse, where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the semi-major and semi-minor axes.
Here, [tex]\( a^2 = 25 \)[/tex], so:
[tex]\[ a = \sqrt{25} = 5 \][/tex]
And [tex]\( b^2 = 9 \)[/tex], so:
[tex]\[ b = \sqrt{9} = 3 \][/tex]
Given these values, the equation:
[tex]\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \][/tex]
describes an ellipse centered at the origin (0,0) with a semi-major axis of length 5 and a semi-minor axis of length 3.
As a more concise representation, our final and simplified equation remains:
[tex]\[ \frac{x^2}{25} + \frac{y^2}{9} - 1 = 0 \][/tex]
This is the exact form representing an ellipse.
The given equation is:
[tex]\[ \frac{x^2}{5^2} + \frac{y^2}{3^2} = 1 \][/tex]
First, we simplify the denominators:
[tex]\[ 5^2 = 25 \quad \text{and} \quad 3^2 = 9 \][/tex]
Inserting these into the equation, we get:
[tex]\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \][/tex]
Next, we rewrite this equation in a more standard form:
[tex]\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \][/tex]
Now, let us transform this equation to make it look like an equation used in identifying conic sections (like ellipses). Notice that an equation of this form:
[tex]\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \][/tex]
is the standard form of an ellipse, where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the semi-major and semi-minor axes.
Here, [tex]\( a^2 = 25 \)[/tex], so:
[tex]\[ a = \sqrt{25} = 5 \][/tex]
And [tex]\( b^2 = 9 \)[/tex], so:
[tex]\[ b = \sqrt{9} = 3 \][/tex]
Given these values, the equation:
[tex]\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \][/tex]
describes an ellipse centered at the origin (0,0) with a semi-major axis of length 5 and a semi-minor axis of length 3.
As a more concise representation, our final and simplified equation remains:
[tex]\[ \frac{x^2}{25} + \frac{y^2}{9} - 1 = 0 \][/tex]
This is the exact form representing an ellipse.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.