ELLIPSE with
Center at (h,k)
Group-2
At the end of the learning episode,
you are expected to:
01
Graph an ellipse given an equation in
standard form with center at (h,k)
Like the other conics, we can move the ellipse so
that its axes are not on the 𝒙 − 𝒂𝒙𝒊𝒔 and 𝒚 − 𝒂𝒙𝒊𝒔.
For the horizontal major axis case , if we move the
intersection of the major and minor axes to the
point (h, k), we have the standard equation of an
ellipse:
Standard Equation of an Ellipse
with center at (h,k)
where:
Features of the graph of an
ellipse:
The ellipse is as follows:
Major axis: horizontal
Center: (h, k)
Foci: (𝒉 − 𝒄, 𝒌) & 𝑭₂ (𝒉 + 𝒄, 𝒌)
Vertices: (𝒉 − 𝒂, 𝒌) & 𝑽₂ (𝒉 + 𝒂, 𝒌)
Covertices: (𝒉, 𝒌 − 𝒃) & 𝑾₂ (𝒉, 𝒌 + 𝒃)
Length of minor axis: 2b
Length of major axis: 2a
Length of foci: 2c
For the vertical major axis case , if we move
the intersection of the major and minor axes
to the point (h, k), we have the standard
equation of an ellipse:
where:
Features of the graph of an
ellipse:
The ellipse is as follows:
Major axis: vertical
Center: (h, k)
Foci: (𝒉, 𝒌 − 𝒄) & 𝑭₂ (𝒉, 𝒌 + 𝒄)
Vertices: (𝒉, 𝒌 − 𝒂) & 𝑽₂ (𝒉, 𝒌 + 𝒂)
Covertices: (𝒉 − 𝒃, 𝒌) & 𝑾₂ (𝒉 + 𝒃, 𝒌)
Length of minor axis: 2b
Length of major axis: 2a
Length of foci: 2c
Example 1
Solution:
Find the values of a, b, and c:
Center: (h, k) → (3, 1)
Foci: (𝒉 – c ,) & 𝑭₂ (𝒉 + c, 𝒌)
(3 – 3,1) & (3 + 3, 1)
→ (0, 1) & (6, 1)
Vertices: (𝒉 – a, 𝒌) & 𝑽₂ (𝒉 + a, 𝒌)
(3 – 5,1) & (3 +5, 1)
→ (−2, 1) & (8, 1)
Covertices: (𝒉, 𝒌 – b ) & 𝑾₂ (𝒉, 𝒌 + b)
(3, 1 – 4)& (3, 1 + 4)
→ (3, −3) & (3, 5)
Based on the graph above, the major axis is horizontal
and its length is 10 units while the length of its minor
axis is 8 units. The length of the foci is 6 units.
Example 2
Sketch the ellipse with equation:
Solution: Find the value of a, b, & c.
𝑎² = 16
a=4
𝑏² = 9
𝑏=3
To find c:
Center: (𝒉, 𝒌) → (−𝟏, −𝟒)
Foci: (𝒉, 𝒌 – c) & 𝑭₂ (𝒉, 𝒌 + c)
Vertices: (𝒉, 𝒌 – a) & 𝑽₂ (𝒉, 𝒌 + a)
(−1, −4 – 4)& (−1, −4 + 4)
→ (−𝟏, −𝟖) & (−𝟏, 𝟎)
Covertices: (𝒉 − 𝒃, 𝒌) & 𝑾₂ (𝒉 + 𝒃, 𝒌)
(−1 – 3,−4)& (−1 + 3,−4)
→ (−𝟒, −𝟒) & (𝟐, −𝟒)
Based on the graph above, the major axis is vertical and its
length is 8 units while the length of its minor axis is 6 units.
The length of the foci is 2√7 units, approximately 5.292 units
END
Group 2