Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation 2𝑥2−4𝑥𝑦+5𝑦2=6.

Question

Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation Determine the nature of conic equation 2𝑥2−4𝑥𝑦+5𝑦2=6.

Answer 1

General form of conic sections:

A x² + B x y + C y² + D x + E y + F = 0

Now identifying conic sections by the discriminant.

1.

If A = C then the graph is a circle

2.

If B² - 4 A C < 0 then the graph is an ellipse.

3.

If B² - 4 A C > 0 then the graph is a hyperbola.

4.

If B² - 4 A C = 0 then the graph is a parabola.

In this case;

2 x² - 4 x y + 5 y² = 6

Subtract 6 to both sides:

2 x² - 4 x y + 5 y² - 6 = 0

In equation:

A x² + B x y + C y² + D x + E y + F = 0

coefficients are:

A = 2 , B = - 4 , C = 5 , D = 0 , E = 0 , F = 6

B² - 4 A C = ( - 4 )² - 4 ∙ 2 ∙ 5 = 16 - 40 = - 24 < 0

The graph is an ellipse.

Answer 2

Well, well, well! Looks like you really want to determine the nature of that conic equation! I admire your persistence. Alright, let's get down to business.

The given equation is 2𝑥² - 4𝑥𝑦 + 5𝑦² = 6. To determine the nature of this conic equation, we can analyze its coefficients.

First, let's focus on the coefficients of 𝑥² and 𝑦². The coefficient of 𝑥² is 2, and the coefficient of 𝑦² is 5. Since both coefficients are positive, this tells us that the conic section will either be an ellipse or a circle. Exciting, right?

Now, let's zoom in on the coefficient of 𝑥𝑦, which is -4. Ah, a negative coefficient! This means that the conic cannot be a circle. Goodbye, circle (at least for now)!

With the coefficient analysis complete, we can conclude that the given conic equation is an ellipse. Ta-da! We did it!

Remember, being persistent pays off, just like your determination to determine the nature of the conic equation. Keep up the great work!

Answer 3

The given equation is 2𝑥²−4𝑥𝑦+5𝑦²=6.

To determine the nature of the conic section represented by this equation, we can start by looking at the coefficients of the variables.

There are three coefficients: 2x², -4xy, and 5y². To determine the nature of the conic section, we need to consider the discriminant, which is the determinant of the coefficient matrix.

The coefficient matrix is:

[2 -2]
[-2 5]

The discriminant can be calculated as follows:

Δ = det ([2 -2]
[-2 5])

Δ = (2)(5) - (-2)(-2)
Δ = 10 - 4
Δ = 6

Since the discriminant (Δ) is positive (Δ > 0), we know that the conic section is an ellipse.

Therefore, the nature of the conic equation 2𝑥²−4𝑥𝑦+5𝑦²=6 is an ellipse.

Answer 4

To determine the nature of the conic equation 2𝑥²−4𝑥𝑦+5𝑦²=6, we can follow these steps:

Step 1: Rewrite the equation in standard form.
To determine the nature of the conic equation, it is important to express it in standard form. The standard form for a conic equation with variables x and y is of the form:

𝐴𝑥² + 𝐵𝑥𝑦 + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0

In our case, the equation 2𝑥²−4𝑥𝑦+5𝑦²=6 is already in standard form.

Step 2: Calculate the discriminant (𝐵²−4𝐴𝐶).
The discriminant, which is calculated using the coefficients A, B, and C of the standard form equation, helps to determine the nature of the conic section.

For our equation 2𝑥²−4𝑥𝑦+5𝑦²=6, the coefficients are:
𝐴 = 2, 𝐵 = -4, 𝐶 = 5

Calculating the discriminant:
𝐵² − 4𝐴𝐶 = (-4)² - 4(2)(5) = 16 - 40 = -24

Step 3: Analyze the discriminant to determine the nature of the conic equation.
By analyzing the value of the discriminant, we can determine the type of conic section the equation represents:
1. If the discriminant is positive (i.e., greater than 0), the conic section is an ellipse.
2. If the discriminant is negative (i.e., less than 0), the conic section is a hyperbola.
3. If the discriminant is zero, the conic section is a parabola.

In our case, the discriminant is negative (-24). Therefore, the nature of the conic equation 2𝑥²−4𝑥𝑦+5𝑦²=6 is a hyperbola.