The equation(s) of the tangent(s) to the ellipse 9(x - 1)2 + 4y2 = 36 parallel to the latus rectum, is (are)

To find the equations of the tangents to the ellipse given by $9(x - 1)^2 + 4y^2 = 36$ that are parallel to the latus rectum, we can follow these steps:

1. Break Down the Problem

1. Identify the given ellipse equation and rewrite it in standard form.
2. Understand the characteristics of the latus rectum and how it relates to tangents.
3. Derive the equation of the tangent line parallel to the latus rectum.

2. Relevant Concepts

• The standard form of the ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$
• The latus rectum of an ellipse is defined as the line segments through the foci parallel to the minor axis.

3. Analysis and Detail

1. Rewriting the Ellipse Equation: $9(x - 1)^2 + 4y^2 = 36 \implies \frac{(x - 1)^2}{4} + \frac{y^2}{9} = 1$ Here, $h = 1$, $k = 0$, $a^2 = 4$ (thus $a = 2$), and $b^2 = 9$ (thus $b = 3$).

2. Identifying the Latus Rectum: The latus rectum of this ellipse is vertical (since $b > a$), and its length is given by $\frac{2b^2}{a} = \frac{2 \cdot 9}{2} = 9$.

3. Equation of Tangent Lines: The tangents parallel to the latus rectum will be horizontal lines of the form $y = k$, where $k$ corresponds to the y-coordinates for which the tangents exist on the ellipse.

   To find valid values for $k$, we substitute $y = k$ into the ellipse equation: $9(x - 1)^2 + 4k^2 = 36$ This rearranges to: $9(x - 1)^2 = 36 - 4k^2$ For tangents to exist, the expression on the right must be non-negative: $36 - 4k^2 \geq 0 \implies 4k^2 \leq 36 \implies k^2 \leq 9 \implies -3 \leq k \leq 3$

4. Verify and Summarize

The values $k = -3, -2, -1, 0, 1, 2, 3$ yield the equations of horizontal tangents:

1. $y = 3$
2. $y = 2$
3. $y = 1$
4. $y = 0$
5. $y = -1$
6. $y = -2$
7. $y = -3$

Final Answer

The equations of the tangents to the ellipse $9(x - 1)^2 + 4y^2 = 36$ that are parallel to the latus rectum are:

$y = 3, \quad y = 2, \quad y = 1, \quad y = 0, \quad y = -1, \quad y = -2, \quad y = -3$