What is (3p+2q)2=9p^2+4q^2 ?

The solution to (3p+2q)2=9p^2+4q^2 is Ellipse with (h,k)=(1/3 , 1/2),a=(sqrt(2))/3 ,b=sqrt(1/2)

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(3p+2q)2=9p^2+4q^2

(3 p + 2 q) 2 = 9 p^{2} + 4 q^{2}

(h, k) = (\frac{1}{3}, \frac{1}{2}), a = \frac{2}{3}, b = \frac{1}{2}

Solution steps

(3 p + 2 q) \cdot 2 = 9 p^{2} + 4 q^{2}

Rewrite

(3 p + 2 q) 2 - 9 p^{2} - 4 q^{2} = 0

in the form of the standard ellipse equation

\frac{( p - \frac{1}{3} ) ^{2}}{( \frac{2}{3} ) ^{2}} + \frac{( q - \frac{1}{2} ) ^{2}}{( \frac{1}{2} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (\frac{1}{3}, \frac{1}{2}), a = \frac{2}{3}, b = \frac{1}{2}

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (\frac{1}{3}, \frac{1}{2}), b = \frac{1}{2}, a = \frac{2}{3}

\frac{( x + 2 ) ^{2}}{16} + \frac{( y - 2 ) ^{2}}{36} = 1

\frac{( x + 3 ) ^{2}}{16} + \frac{( y - 4 ) ^{2}}{9} = 1

\frac{( x - 5 ) ^{2}}{9} + \frac{( y - 4 ) ^{2}}{36} = 1

\frac{x ^{2}}{4} + \frac{y ^{2}}{20} = 1

(x - 1)^{2} + \frac{( y + 2 ) ^{2}}{4} = 1

What is (3p+2q)2=9p^2+4q^2 ?
The solution to (3p+2q)2=9p^2+4q^2 is Ellipse with (h,k)=(1/3 , 1/2),a=(sqrt(2))/3 ,b=sqrt(1/2)