Classify the conic represented by the following equation: [tex]2x^2-10x+7y^2+y+8=0[/tex]
Circle
Ellipse
Parabola
Hyperbola
Answer (1)
Complete the square for the x terms: 2 x 2 − 10 x A rr 2 ( x − 2 5 ) 2 − 2 25 .
Complete the square for the y terms: 7 y 2 + y A rr 7 ( y + 14 1 ) 2 − 28 1 .
Substitute back into the original equation and simplify: 2 ( x − 2 5 ) 2 + 7 ( y + 14 1 ) 2 = 28 127 .
Since the coefficients of the x 2 and y 2 terms are both positive and different, the conic section is an e ll i p se .
Explanation
Analyze the equation We are given the equation 2 x 2 − 10 x + 7 y 2 + y + 8 = 0 . Our goal is to classify the conic section that this equation represents. The possible classifications are circle, ellipse, parabola, and hyperbola. To do this, we will rewrite the equation in a standard form by completing the square for both x and y terms.
Complete the square for x terms First, let's complete the square for the x terms: 2 x 2 − 10 x . We can factor out the 2 to get 2 ( x 2 − 5 x ) . To complete the square, we need to add and subtract ( 2 5 ) 2 = 4 25 inside the parenthesis. So we have:
2 ( x 2 − 5 x + 4 25 − 4 25 ) = 2 (( x − 2 5 ) 2 − 4 25 ) = 2 ( x − 2 5 ) 2 − 2 25 .
Complete the square for y terms Next, let's complete the square for the y terms: 7 y 2 + y . We can factor out the 7 to get 7 ( y 2 + 7 1 y ) . To complete the square, we need to add and subtract ( 14 1 ) 2 = 196 1 inside the parenthesis. So we have:
7 ( y 2 + 7 1 y + 196 1 − 196 1 ) = 7 (( y + 14 1 ) 2 − 196 1 ) = 7 ( y + 14 1 ) 2 − 28 1 .
Substitute back into original equation Now, substitute these back into the original equation:
2 ( x − 2 5 ) 2 − 2 25 + 7 ( y + 14 1 ) 2 − 28 1 + 8 = 0
2 ( x − 2 5 ) 2 + 7 ( y + 14 1 ) 2 = 2 25 + 28 1 − 8 = 28 350 + 28 1 − 28 224 = 28 127
Classify the conic section So the equation becomes:
2 ( x − 2 5 ) 2 + 7 ( y + 14 1 ) 2 = 28 127
Since the coefficients of the x 2 and y 2 terms are both positive and different, this equation represents an ellipse.
Final Answer Therefore, the conic section represented by the given equation is an ellipse.
Examples
Ellipses are commonly encountered in various real-world scenarios. For instance, the orbits of planets around the sun are elliptical, with the sun at one focus. Similarly, elliptical gears are used in machinery to produce variable speed or torque. Understanding the properties of ellipses, such as their equation and shape, is crucial in fields like astronomy, engineering, and architecture for designing and analyzing systems involving elliptical paths or shapes. This problem demonstrates how to classify a conic section, which is a fundamental concept in analytic geometry with practical applications in diverse fields.
