Question

write the equation of the ellipse in standard form, and identify the endpoints of the major and minor axes as well as the foci.
16x² + 64x + 4y² - 8y + 4 = 0
equation in standard form:
endpoints of major axis: ( , ), ( , )
endpoints of minor axis: ( , ), ( , )
foci are at ( , ), ( , )
question help: video ebook

Explanation:

Step1: Complete the square for \(x\) and \(y\) terms

Given equation \(16x^{2}+64x + 4y^{2}-8y+4 = 0\).
Group \(x\) - terms and \(y\) - terms:
\(16(x^{2}+4x)+4(y^{2}-2y)+4 = 0\).
For \(x^{2}+4x\), \((x + 2)^{2}=x^{2}+4x + 4\), so \(x^{2}+4x=(x + 2)^{2}-4\).
For \(y^{2}-2y\), \((y - 1)^{2}=y^{2}-2y+1\), so \(y^{2}-2y=(y - 1)^{2}-1\).
Substitute back:
\(16((x + 2)^{2}-4)+4((y - 1)^{2}-1)+4 = 0\).
Expand: \(16(x + 2)^{2}-64+4(y - 1)^{2}-4 + 4 = 0\).
Simplify: \(16(x + 2)^{2}+4(y - 1)^{2}=64\).
Divide by \(64\): \(\frac{(x + 2)^{2}}{4}+\frac{(y - 1)^{2}}{16}=1\).

Step2: Identify \(a\), \(b\), \(h\), \(k\)

For the standard form of an ellipse \(\frac{(x - h)^{2}}{b^{2}}+\frac{(y - k)^{2}}{a^{2}}=1\) (\(a>b>0\)), here \(h=-2\), \(k = 1\), \(a = 4\), \(b = 2\).

Step3: Find endpoints of major and minor axes

For major axis (parallel to \(y\) - axis since \(a>b\)):
Endpoints of major axis: \((h,k\pm a)\), so \((-2,1 + 4)=(-2,5)\) and \((-2,1-4)=(-2,-3)\).
For minor axis: \((h\pm b,k)\), so \((-2+2,1)=(0,1)\) and \((-2 - 2,1)=(-4,1)\).

Step4: Find \(c\) and foci

Use the formula \(c=\sqrt{a^{2}-b^{2}}\), \(c=\sqrt{16 - 4}=\sqrt{12}=2\sqrt{3}\).
Foci: \((h,k\pm c)\), so \((-2,1+2\sqrt{3})\) and \((-2,1 - 2\sqrt{3})\).

Answer:

Equation in standard form: \(\frac{(x + 2)^{2}}{4}+\frac{(y - 1)^{2}}{16}=1\).
Endpoints of major axis: \((-2,5)\), \((-2,-3)\).
Endpoints of minor axis: \((0,1)\), \((-4,1)\).
Foci: \((-2,1 + 2\sqrt{3})\), \((-2,1-2\sqrt{3})\).