Question

how do you solve this equation and determine the conic section: 81x^2 + 4y^2 -324 = 0

Answer 1

Given equation is 81x^2+4y^2-324 = 0

Divide to each side by 324.

81x^2/324+4y^2/324 -324/324 = 0

(x^2/4+y^2/81)-1 = 0

Add 1 to each side.

(x^2/4+y^2/81)-1+1 = 0+1

x^2/2^2+y^2/9^2 = 1

(x/2)^2+ (y/9)^2 = 1

Compare the ellipse equation (x/a)^2+(y/b)^2 = 1

Here a = 2,b = 9.

Given equation is ellipse equation.

Answer 2

The diophantine equation

The term Diophantine Equation means that the solutions (x , y ) should be integer numbers.

Compare the above equation .

So the above equation is elliptical.

Elliptical case

Since the ellipse is a closed figure, the number of solutions will be finite.

 

Answer 3

Continuous...

For any value of x  there will be two values of y  except at the left and right extremes of the ellipse. In this case there will be only one value of y . To determine the location of the left and right extremes we should equal the square root to zero, so the previous expression returns only one value of y .

The values of x  should be between the roots of

Substitute the value of x  in 

Substitute x  = 0 in 

Solution

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