a parabola has a focus of (-3,4) and a directrix with equation: 3x-4y-25. Since the distance from the focus to a point (x,y) on the parabola is : squareroot (x+3)^2 +(7-4)^2 and is equal to the distance from the point (x.y) to the line which is 16x^2 + 9y^2 +300x - 400y +24xy=0. do the arithmetic to rotate this curve into recognizable form of a parabolawith axis of symmetry either veritical or horizontal.
First of all, I agree with your equation.
This is a pretty advanced topic for this forum, I will get you going, but will not be able to type the necessary matrices. It does not allow the proper formatting and doesn't line up.
Have you come across this:
The xy term can be eliminated from the equation
ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0
by a rotation through an angle ß, where
tan 2ß = 2h/(b-a)
so in your case of
16x^2 + 9y^2 + 300x - 400y + 24xy = 0
a=16
b=9
h=12
tan 2ß = 24/(9-16) = -24/7
(for reference angle:
2ß= 73.7º)
since the tangent is negative, 2ß lies in the second or fourth quadrants.
Let 2ß lie in the second quadrant so that 2ß = 180-73.7 = 106.3º
so ß = 53.15
So your required rotation matrix is
│cos 53.15 sin 53.15│
│-sin53.15 cos 53.15│
Does any of this make sense ?
To rotate the given curve into the recognizable form of a parabola with an axis of symmetry, we need to perform the following steps:
Step 1: Simplify the given equation.
Step 2: Complete the square to express the equation in standard form.
Step 3: Identify the axis of symmetry.
Step 4: Rewrite the equation in the recognizable form of a parabola.
Let's go through each step in detail:
Step 1: Simplify the given equation.
The equation of the curve is given as: 16x^2 + 9y^2 + 300x - 400y + 24xy = 0
Step 2: Complete the square to express the equation in standard form.
To complete the square, we need to group the x-terms, y-terms, and constant terms separately:
16x^2 + 24xy + 9y^2 + 300x - 400y = 0
Now, let's focus on completing the square for the x-terms:
16x^2 + 24xy + 300x = 0
16(x^2 + (3/2)x) + 300x = 0
To complete the square, we need to take half of the coefficient of the x-term, square it, and add it to both sides:
16(x^2 + (3/2)x + (3/4)^2) + 300x = 16(3/4)^2
Simplifying further:
16(x + 3/4)^2 + 300x = 16(9/16)
16(x + 3/4)^2 + 300x = 9
Now, let's complete the square for the y-terms:
9y^2 - 400y = 0
9(y^2 - (400/9)y) = 0
Taking half of the coefficient of the y-term, squaring it, and adding it to both sides:
9(y^2 - (400/9)y + (200/9)^2) = 9(200/9)^2
9(y - 400/9)^2 = 400/9
Step 3: Identify the axis of symmetry.
The axis of symmetry for a parabola can be identified by the term in parentheses of each squared term. In this case, the x-term is (x + 3/4) and the y-term is (y - 400/9). The axis of symmetry is parallel to either the x-axis or the y-axis, depending on which term is affected by the squared term.
Since the x-term is squared, the axis of symmetry is vertical, parallel to the y-axis.
Step 4: Rewrite the equation in the recognizable form of a parabola.
Now that we have the equation in standard form and identified the axis of symmetry, we can rewrite it in the recognizable form of a parabola.
16(x + 3/4)^2 + 300x + 9(y - 400/9)^2 = 9
Dividing through by 9 to simplify the equation:
(16/9)(x + 3/4)^2 + (300/9)x + (9/9)(y - 400/9)^2 = 1
(16/9)(x + 3/4)^2 + (100/3)x + (1)(y - 400/9)^2 = 1
The equation of the rotated parabola, with the axis of symmetry being vertical, is:
(16/9)(x + 3/4)^2 + (100/3)x + (y - 400/9)^2 = 1
This equation represents a parabola with its vertex at the point (-3/4, 400/9), a vertical axis of symmetry, and a recognizable form.