Question

How do you write the equation for an ellipse with centre (5,-4), vertical major axis of length 12 and minor axis of length 8?

Answer 1

Hint: In the above given question you need to know the general equation of ellipse, which is described as vertical major axis(ellipse is defined for both vertical and horizontal major axis), and then by using the values as per the question we can simply write the equation for ellipse.

Formula Used:
General equation of ellipse with having vertical major axis:
\[ \dfrac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} + \dfrac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1\]

Complete step by step solution:
The given question is about writing an equation for ellipse which have vertical major axis, here in the solution first we need to have the general equation of the ellipse which have vertical axis as major axis, we get the equation of ellipse as;
\[ \dfrac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} + \dfrac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1\]
Here,
(h,k)= centre of the ellipse
2a= major axis
2b=minor axis
Now as per the data given in the question:
We have,
\[
   \Rightarrow (h,k) = (5, - 4) \\
   \Rightarrow 2a = 12 \to a = 6 \\
   \Rightarrow 2b = 8 \to b = 4 \\
 \]
Putting this values in the general equation of ellipse with major axis as vertical, we get the equation as:
\[
   \Rightarrow \dfrac{{{{\left( {x - 5} \right)}^2}}}{{{{(6)}^2}}} + \dfrac{{{{\left( {y - ( - 4)} \right)}^2}}}{{{{(4)}^2}}} = 1 \\
   \Rightarrow \dfrac{{{{\left( {x - 5} \right)}^2}}}{{36}} + \dfrac{{{{\left( {y + 4} \right)}^2}}}{{16}} = 1 \\
 \]
Hence our required equation of ellipse is obtained with the major axis as vertical.

Note: The given question is of coordinate geometry in which we know the equations of some standard shapes, like circle, ellipse, parabola etc. here we need to obtain the equation or we need find any point or coordinates and the conditions are provided in the question, to solve these problems one should know at least property and general equations of the desired figures or shape.