What is (5p+3q)2=25p^2+9q^2 ?

The solution to (5p+3q)2=25p^2+9q^2 is Ellipse with (h,k)=(1/5 , 1/3),a=(sqrt(2))/5 ,b=(sqrt(2))/3

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(5p+3q)2=25p^2+9q^2

(5 p + 3 q) 2 = 25 p^{2} + 9 q^{2}

(h, k) = (\frac{1}{5}, \frac{1}{3}), a = \frac{2}{5}, b = \frac{2}{3}

Solution steps

(5 p + 3 q) \cdot 2 = 25 p^{2} + 9 q^{2}

Rewrite

(5 p + 3 q) 2 - 25 p^{2} - 9 q^{2} = 0

in the form of the standard ellipse equation

\frac{( p - \frac{1}{5} ) ^{2}}{( \frac{2}{5} ) ^{2}} + \frac{( q - \frac{1}{3} ) ^{2}}{( \frac{2}{3} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (\frac{1}{5}, \frac{1}{3}), a = \frac{2}{5}, b = \frac{2}{3}

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (\frac{1}{5}, \frac{1}{3}), b = \frac{2}{3}, a = \frac{2}{5}

25 x^{2} + 9 y^{2} + 100 x + 36 y - 89 = 0

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

\frac{( x - 12 ) ^{2}}{100} + \frac{( y + 7 ) ^{2}}{36} = 1

12 x^{2} + 10 y^{2} = 120

25 x^{2} + 9 y^{2} - 50 x + 54 y - 119 = 0

What is (5p+3q)2=25p^2+9q^2 ?
The solution to (5p+3q)2=25p^2+9q^2 is Ellipse with (h,k)=(1/5 , 1/3),a=(sqrt(2))/5 ,b=(sqrt(2))/3