Question

Find the major axis, minor axis, centre and eccentricity of the ellipse 4(x-2y+1)^2+9(2x+y+2)^2=180

Answer 1

Answer:

Step-by-step explanation:

Given to find the major axis, minor axis, centre and eccentricity of the ellipse 4(x-2y+1)^2+9(2x+y+2)^2=180

Equation of major axis is 2x + y + 2=0 and equation of minor axis is x -2y + 1 = 0

4(x - 2y + 1)^2 + 9(2x + y + 2)^2 = 180

Dividing by 180 we get

(x - 2y + 1/ 45)^2 + (2x + y + 2/20)^2 = 1  

we have a1, b1 and a2,b2

√a1^2 + b1^2 = √1+4 = √5

√a2^ + b2^2 = √4 + 1=√5

So from the equation we get

(x - 2y + 1 / √5)^2/45/5 + (2x + y + 2/√5)^2/20/5

Now we get the standard form

(x - 2y + 1/√5)^2/9 + (2x + y + 2/√5)^2/4 = 1

 Now a^2 = 9 or a = 3

          b^2 = 4 or b = 2

Now  major axis is 2 a = 2(3) = 6

        minor axis is 2 b = 2(2) = 4

eccentricity = c/a        

       e = c/a

     a^2 - b^2 = c^2

   9 - 4 = c^2

c = √5

Now e = √5 / 3

Centre is (-1,0)