Q.Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $16x^2 + y^2 = 16$.

Concept: Standard form of an ellipse and its geometric parameters.

Divide both sides by 16 to get the standard form:

This is $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ with $a^2 = 16$ and $b^2 = 1$, so $a = 4$ and $b = 1$. Since $a > b$, the major axis lies along the $y$-axis.

For an ellipse, $c^2 = a^2 - b^2 = 16 - 1 = 15$, giving $c = \sqrt{15}$.

Now we extract all parameters:

• Vertices (endpoints of major axis): $(0, \pm a) = (0, \pm 4)$
• Foci: $(0, \pm c) = (0, \pm\sqrt{15})$
• Length of major axis: $2a = 8$
• Length of minor axis: $2b = 2$ …

The equation $16x^2 + y^2 = 16$ describes an ellipse, but not in the form we can immediately read. An ellipse in standard position has the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where the denominators tell us the semi-axis lengths. The larger denominator corresponds to the major axis. Once we identify $a$ (semi-major) and $b$ (semi-minor), the geometry follows: the foci lie along the major axis at a distance $c = \sqrt{a^2 - b^2}$ from the center, the eccentricity is $e = \frac{c}{a}$, and the latus rectum (the chord through a focus perpendicular to the major axis) has length $\frac{2b^2}{a}$.

Let's extract everything systematically.

Step-by-step solution

1. Rewrite in standard form Divide both sides of $16x^2 + y^2 = 16$ by $16$:

This is now in the form $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ with $b^2 = 1$ and $a^2 = 16$.

2. Identify the semi-axes

   Since $a^2 = 16 > b^2 = 1$, the major axis lies along the $y$-direction.

   We have $a = 4$ (semi-major axis) and $b = 1$ (semi-minor axis).

3. Find the distance to the foci

   For an ellipse, $c^2 = a^2 - b^2$:

The foci lie on the major axis, which is the $y$-axis, at $(0, \pm c)$.

4. Locate the vertices

   The vertices are the endpoints of the major axis, at $(0, \pm a) = (0, \pm 4)$.

5. Compute the lengths of the axes

   • Major axis: $2a = 2 \times 4 = 8$.
   • Minor axis: $2b = 2 \times 1 = 2$.

6. Calculate the eccentricity

   The eccentricity measures how "stretched" the ellipse is:

7. Find the length of the latus rectum …