Obtain the Differential Equation from the family of ellipses centered

zgribestika

zgribestika

Answered question

2021-12-28

Obtain the Differential Equation from the family of ellipses centered at (h,k) whose semi-major axis is twice the semi-minor axis.

Answer & Explanation

xandir307dc

xandir307dc

Beginner2021-12-29Added 35 answers

Equation from family of ellipses whose centered at (h, k) and semi mayor axis is twice the semi minor axis
mayor axis =a
minor axis =b
General equation of ellipse:
(xh)2a2+(yk)2b2=1 (1)
Given condition:
Semi mayor axis is twice the semi minor
a2=2(b2)a2=b (put in 1)
Then 1 becomes (xh)2a2+(yk)2(a2)2=1
(xh)2a2+4(yk)2a2=1
(xh)2+4(yk)2=a2
2(xh)+g(yk)dydx=0
dydx=2(x+h)8(yk)(xh)4(yk)=dydx
servidopolisxv

servidopolisxv

Beginner2021-12-30Added 27 answers

Step 1
Given data is:
The semi-major axis is twice the semi-minor axis and the center of the ellipse is (h,k)
To find the differential equation from the family of the ellipse.
Step 2
Let a be the major axis and b is the minor-axis
The center of the ellipse is (h,k), therefore the equation of the ellipse is:
(xh)2a2+(yk)2b2=1 (1)
From the question,
The semi-major axis is twice the semi-minor axis, therefore
a=2b
From equation (1),
(xh)2(2b)2+(yk)2b2=1
(xh)24b2+(yk)2b2=1
(xh)2+4(yk)2=4b2 (2)
Step 3
Differentiate equation (2) with respect to x,
2(xh)+4×2(yk)y=0
2(xh)+8(yk)y=0
Again differentiate with respect to x:
2×1+8{(yk)yy×y}=0
8(yk)y8(y)2=1
(yk)y(y)2=18
Thus, the differential equation is (yk)y(y)2=18
user_27qwe

user_27qwe

Skilled2022-01-05Added 375 answers

\(\begin{array}{}\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 (1) \a=2b \\frac{(x-h)^{2}}{(2b)^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \\frac{(x-h)^{2}}{4b^{2}}+\frac{(y-k)^{2}}{b^{2}}=1 \(x-h)^{2}+4(y-k)^{2}=4b^{2} (2) \2(x-h)+4 \times 2(y-k)y=0

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