Question

5x2+9y2-12xy-10x+25=0

Answer 1

В приложении .................

Answer 2

The given equation is:

5x^2 + 9y^2 - 12xy - 10x + 25 = 0

This equation represents a conic section, specifically an ellipse. To determine the properties of the ellipse, we can rewrite the equation in a standard form, which is:

[(x - h)^2 / a^2] + [(y - k)^2 / b^2] = 1

where (h, k) represents the center of the ellipse, and 'a' and 'b' represent the semi-major and semi-minor axes lengths, respectively.

To simplify the given equation, we can complete the square for both 'x' and 'y' terms. Let's do that step by step:

1. Group the terms involving 'x' and 'y' together: 5x^2 - 12xy - 10x + 9y^2 + 25 = 0

2. Rearrange the terms: 5x^2 - 12xy - 10x + 9y^2 = -25

3. Factor out 5 from the 'x' terms and 9 from the 'y' terms: 5(x^2 - 2xy) - 10x + 9(y^2) = -25

4. Complete the square for the 'x' terms: 5(x^2 - 2xy + (y^2/4)) - 10x + 9(y^2) = -25 + 5(y^2/4)

5. Complete the square for the 'y' terms: 5(x^2 - 2xy + (y^2/4)) + 9(y^2 - (10/9)y) = -25 + 5(y^2/4) + 9(y^2 - (10/9)y)

6. Simplify the right-hand side of the equation: 5(x^2 - 2xy + (y^2/4)) + 9(y^2 - (10/9)y) = -25 + (5/4)y^2 + 9y^2 - (50/9)y

Combining like terms, we have: 5(x^2 - 2xy + (y^2/4)) + 9(y^2 - (10/9)y) = (29/4)y^2 - (50/9)y - 25

7. Now, we need to determine the values of 'h', 'k', 'a', and 'b'. To do that, we'll complete the square for both 'x' and 'y' terms inside the parentheses.

For the 'x' terms: x^2 - 2xy + (y^2/4) = (x - y/2)^2

For the 'y' terms: y^2 - (10/9)y = (y - 5/9)^2 - 25/81

Substituting these results back into the equation, we get: 5(x - y/2)^2 + 9[(y - 5/9)^2 - 25/81] = (29/4)y^2 - (50/9)y - 25

8. Distribute and simplify: 5(x - y/2)^2 + 9(y - 5/9)^2 - 9(25/81) = (29/4)y^2 - (50/9)y - 25

Multiplying through by 36 (the least common multiple of 4 and 9) to eliminate the denominators, we have: 180(x - y/2)^2 + 324(y -