Questions: Write the equation of the ellipse (16 x^2+25 y^2+160 x-50 y+25=0) in standard form (frac(x-h)^2p^2+frac(y-k)^2q^2=1) Where: (h=) (k=) (p=) (q=)

Solution Steps

To convert the given ellipse equation into standard form, we need to complete the square for both the \(x\) and \(y\) terms. This involves rearranging and grouping the terms, completing the square for each variable, and then simplifying the equation to match the standard form of an ellipse.

The given equation of the ellipse is

\[ 16x^2 + 25y^2 + 160x - 50y + 25 = 0. \]

We focus on the \(x\) terms:

\[ 16x^2 + 160x. \]

Factoring out 16 gives us

\[ 16(x^2 + 10x). \]

To complete the square, we take half of 10, square it, and add/subtract it inside the parentheses:

\[ 16\left((x + 5)^2 - 25\right) = 16(x + 5)^2 - 400. \]

Next, we handle the \(y\) terms:

\[ 25y^2 - 50y. \]

Factoring out 25 gives us

\[ 25(y^2 - 2y). \]

Completing the square involves taking half of -2, squaring it, and adding/subtracting:

\[ 25\left((y - 1)^2 - 1\right) = 25(y - 1)^2 - 25. \]

Substituting the completed squares back into the original equation, we have:

\[ 16(x + 5)^2 - 400 + 25(y - 1)^2 - 25 + 25 = 0. \]

This simplifies to:

\[ 16(x + 5)^2 + 25(y - 1)^2 - 400 = 0. \]

Rearranging gives us:

\[ 16(x + 5)^2 + 25(y - 1)^2 = 400. \]

Dividing through by 400 to get the standard form:

\[ \frac{(x + 5)^2}{25} + \frac{(y - 1)^2}{16} = 1. \]

From the standard form \(\frac{(x - h)^2}{p^2} + \frac{(y - k)^2}{q^2} = 1\), we identify:

• \(h = -5\)
• \(k = 1\)
• \(p = 5\)
• \(q = 4\)

Final Answer