Ellipse Problem? Please help.?

Question

Find the equation for the ellipse if it passes through (8,6), centered at the origin, and length of major axis is 32 on the x-axis. Meaning 16 on the positive x-axis and 16 on the negative x-axis.

Answer 1

There are two ways to write the general formula for an ellipse. Since your ellipse is centered on the origin, that simplifies the general formula.

One way to write it is

x²/a² + y²/b² = 1

where a is the semi along the x-axis and b is the semi along the y-axis.

So we can say

x²/16² + y²/b² = 1

We have to find b.

Wal's method for finding b looks pretty good.

Answer 2

The general equation of an ellipse centered at the origin is (x^2)/(a^2) + (y^2)/(b^2) = 1. Le length of the major axis is 2a = 32, i.e. a = 16, since it passes through the point (8,6), one has (8^2)/(16^2) + (6^2)/(b^2) =1 which implies b = 4sqrt(3). Thus the equation should be (x^2)/(16^2) + (y^2)/(3*16) = 1.

Answer 3

Kevin had a brilliant and the final option answer. the only difficulty i could co in yet in any different case is the quantity cruching, which could be simplified. With center on the beginning place and the critical axes on the x-axis and y-axis, then the equation for the ellipse is: x²/m² + y²/n² = one million it somewhat is the final style for the ellipse. on the grounds that (x,y) it passes with the aid of (2,2) and (3,-one million), then you definately can substitute them to the equation. (2)²/m² + (2)²/n² = one million (3)²/m² + (-one million)²/n² = one million as a result, 4/m² + 4/n² = one million 9/m² + one million/n² = one million We multiply with the aid of m²n² 4m² + 4n² = m²n² m² + 9n² = m²n² on the grounds that the two equations = m²n², set them equivalent. 4m² + 4n² = m² + 9n² 3m² = 5n² m² = (5/3)n² on the grounds that m and n are continuously going for use while they are squared, we don't would desire to take the sq. root. Plugging the 1st element into the equation we've: 4/[(5/3)n²] + 4/n² = one million Multiplying with the aid of n² 4/[(5/3)] + 4 = n² Grouping words 4[ 3/5 +one million] = n² 4(8/5) = n² n² = 32/5 m² = (5/3)n² m² = (5/3)(32/5) = 32/3 as a result, the equation of the ellipse is: x²/(32/3) + y²/(32/5) = one million Or, 3x²/32 + 5y²/32 = one million Or, 3x² + 5y² = 32

Answer 4

ellipse of the form
x²/a² + y²/b² = 1

a = 16 and (8, 6) lies on the ellipse.

So 8²/16² + 6²/b² = 1
ie 1/4 + 36/b² = 1
ie 36/b² = 3/4
ie b²/36 = 4/3
b² = 144/3 = 48
b = 4√3

So the ellipse is
x²/16² + y²/(4√3)² = 1

Answer 5

In the standard form for a centered ellipse (above) you already know a, so getting b is just basic algebra:

b² = 6²/(1-8²/16²)