# How do you put #6x^2+5y^2-24x+20y+14=0# in standard form, find the center, the endpoints, vertices, the foci and eccentricity?

##### 1 Answer

Standard forms for the equation of an ellipse are

#### Explanation:

Reference for an ellipse

Given:

Add

Remove a common factor of 6 from the first 3 terms and a common factor of 5 from the next 3 terms:

To find the value of h, set the middle term of the right side of the pattern

Substitute the left side of the pattern into equation [1]:

Substitute 2 for h everywhere in equation [2]:

To find the value of k, set the middle term of the right side of the pattern

Substitute the left side of the pattern into equation [3]:

Substitute -2 for k everywhere in equation [4]:

Simplify the right side of equation [5]:

Divide both sides of the equation by 30:

Make the denominators squares and swap terms:

Equation [8] is the standard form where,

From the reference:

The center is at

The vertices are at #(h, k-a) and (h,k+a):

The foci are at: #(h, k-c) and (h,k+c):

The co-vertices are at

The eccentricity is