X^2+3y^2-4x+24y+3=0 i know it is ellipse.

Question

but, i dont know how to find vertex and focus or all the important points.?

Answer 1

The ellipse equation

Find the stadard form equation using completing square.

To change the expressions (x ²- 4x) and (y ² + 8y) into a perfect square trinomials,

add (half the x coefficient)² and add (half the y coefficient)²to each side of the equation.

The standard form for an ellipse is in a form = 1, So divide both sides of equation by 49 to set it equal to 1.

Compare it to standard form of ellipse

a ² > b ²

If the larger denominator is under the "x " term, then the ellipse is horizontal.

Center (h, k ) = (2, -4)

a  = length of semi-major axis = 7

b  = length of semi-minor axis = 7/√3.

Vertices (h + a, k )(h - a , k )

Co-vertices (h, k + b )(h , k - b )

c  is distnace from the center to each focus.