A parabola has equation x2 + y2 + 2xy - 6x - 2y + 3=0.how to find the focus from this equation? Please help.

To find the focus of the parabola given by the equation x² + y² + 2xy - 6x - 2y + 3 = 0, we first need to rearrange the equation into a more recognizable form. The presence of the xy term suggests that this is not a standard parabola aligned with the axes, but rather one that has been rotated. Let's break down the steps to determine the focus.

Step 1: Identify the Type of Conic Section

The first step is to rewrite the equation in a standard form. Since the equation contains both x² and y², we have a conic section that may be rotated. We can use the discriminant of the conic section to confirm this. The general form of a conic is Ax² + Bxy + Cy² + Dx + Ey + F = 0. Here, A = 1, B = 2, C = 1, D = -6, E = -2, and F = 3.

• The discriminant is given by B² - 4AC.
• Calculating this gives us: 2² - 4(1)(1) = 4 - 4 = 0.

A discriminant of 0 indicates that this conic is a parabola.

Step 2: Rotating the Axes

Next, we need to eliminate the xy term to rewrite the equation in a standard form. We can achieve this by applying a rotation of axes. The angle of rotation θ that eliminates the xy term can be found using the formula:

tan(2θ) = B / (A - C)

Plugging in our values:

tan(2θ) = 2 / (1 - 1) = 2 / 0

This means θ = 45 degrees or π/4 radians, since the tangent is undefined at this angle.

Step 3: Applying the Rotation

Using the rotation formulas:

• x = x'cos(θ) - y'sin(θ)
• y = x'sin(θ) + y'cos(θ)

Substituting θ = π/4, we get:

• x = (x' - y')/√2
• y = (x' + y')/√2

Substituting these into the original equation will allow us to express the equation in terms of x' and y'. This will eliminate the xy term.

Step 4: Simplifying the Equation

After substitution and simplification, we will arrive at a new equation in the form of a parabola, typically x'² = 4py' for horizontal parabolas or y'² = 4px' for vertical ones. From this new equation, we can determine the vertex and the focus.

Locating the Focus

In a standard parabola x'² = 4py', the focus is located at (0, p). If we find that our equation simplifies to this form, we can easily identify p. The coordinates of the focus will directly depend on the value of p, which represents the distance from the vertex to the focus along the axis of symmetry.

Final Steps

After following these steps, you will have the coordinates of the focus based on the calculations and simplifications made. It's essential to keep track of the transformations you've applied during this process, as they will influence the final position of the focus in the original coordinate system.

By following these logical steps, you can successfully determine the focus of the parabola from the given equation. If you need any further clarification on any of the steps, feel free to ask!