Identify the graph of 4y^(2)+12x^(2)=24 for T(-1,4) and write an equation of the translated or rotated graph in general form.

The point T(-1,4) lies on the ellipse on the graph.

The equation of the ellipse in generalform is:

(x/√2)² + (y/√6)² = 1

What is an equation?

An equation contains one or more terms with variables connected by an equal sign.

Example:

2x + 4y = 9 is an equation.

2x = 8 is an equation.

We have,

The given equation, 4y² + 12x² = 24, can be simplified by dividing both sides by 24 to get:

y²/6 + x²/2 = 1

This is the equation of an ellipse centered at the origin, with a horizontal majoraxis of length 2√6 and a vertical minoraxis of length 2√2.

To identify the graph of the ellipse passing through T(-1,4), we substitute x = -1 and y = 4 in the equation and simplify:

4(4)² + 12(-1)² = 24

64 - 12 = 52

Thus, point T(-1,4) lies on the ellipse.

To find the equation of the translated or rotated ellipse in general form, we can use standard transformations.

Let (h,k) be the center of the ellipse.

Then we have:

(x-h)²/a² + (y-k)²/b² = 1

where a and b are the lengths of the semi-major and semi-minor axes, respectively.

The center of the given ellipse is (0,0), so h = 0 and k = 0.

To determine the values of a and b, we note that a^(2) is the coefficient of x² in the equation, and b² is the coefficient of y².

Thus, we have:

a² = 2 and b² = 6

Taking squareroots, we get:

a = √2 and b = √6

Therefore,

The equation of the ellipse in generalform is:

(x/√2)² + (y/√6)² = 1

This is the equation of an ellipse with a horizontal majoraxis of length 2√2 and a vertical minoraxis of length 2√6, centered at the origin and rotated counterclockwise by an angle of θ such that tan(2θ) = 4/3.

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Answer:

b. ellipse; 3x^2 + y^2 + 6x - 8y + 13 = 0

Step-by-step explanation:

thats the answer on edg :)