Find the equation of the ellipse with the centre (3,1) ,vertex (3,-2) and passing through (8,0)

Sure, let's find the equation of the ellipse step by step.

Step 1: Identify the given points The center of the ellipse (h, k) is given as (3, 1). One of the vertices is given as (3, -2). The ellipse also passes through the point (8, 0).

Step 2: Find the semi-major axis length (a) The semi-major axis length is the distance from the center of the ellipse to any of its vertices. We can calculate this using the distance formula:

a = sqrt[(x2 - x1)^2 + (y2 - y1)^2] a = sqrt[(3 - 3)^2 + (-2 - 1)^2] a = sqrt[0 + 9] a = 3

Step 3: Find the semi-minor axis length (b) We know that the ellipse passes through the point (8, 0). The distance from the center of the ellipse to this point is the length of the semi-minor axis.

b = sqrt[(x2 - x1)^2 + (y2 - y1)^2] b = sqrt[(8 - 3)^2 + (0 - 1)^2] b = sqrt[25 + 1] b = sqrt[26]

Step 4: Write the equation of the ellipse The general form of the equation of an ellipse with center at (h, k) and semi-major axis a and semi-minor axis b is:

(x - h)^2/a^2 + (y - k)^2/b^2 = 1

Substituting the values we found:

(x - 3)^2/3^2 + (y - 1)^2/26 = 1

This simplifies to:

(x - 3)^2/9 + (y - 1)^2/26 = 1

So, the equation of the ellipse is (x - 3)^2/9 + (y - 1)^2/26 = 1.