Let's solve the questions one by one based on the provided geometry.

Given Information

1. The circle has center $B(-1, 1)$ and radius $\sqrt{20}$.
2. $BC$ is parallel to the $y$-axis, and $CB = 5$.
3. $\triangle ABC$ and $\triangle ADO$ have angles $\theta$ as specified.

Now, let's go through each question step-by-step.

Question 6.1: Determine the coordinates of $C$.

Since $BC$ is parallel to the $y$-axis and $CB = 5$, the point $C$ lies vertically above or below $B$.

1. Coordinates of $B$ are $(-1, 1)$.
2. Since $BC$ is a vertical line, $C$ will have the same $x$-coordinate as $B$, so $x = -1$.
3. As $CB = 5$, $C$ will be at $( -1, 1 + 5) = (-1, 6)$ or $(-1, 1 - 5) = (-1, -4)$.

From the diagram, $C$ is above $B$, so the coordinates of $C$ are: $C(-1, 6)$

Question 6.2: Calculate the length of $CA$.

To find $CA$, we need the coordinates of $A$, which we will determine later. For now, let’s proceed with known quantities until we can find $A$.

Question 6.3: Write down the value of $\tan \theta$.

The angle $\theta$ is between the line $BC$ (vertical) and $AB$. We can express $\tan \theta$ by determining the slope of $AB$ once we know $A$.

Question 6.4: Show that the gradient of $AB$ is $-2$.

Since $A$ lies on the tangent to the circle at $C$, the slope of $AB$ can be derived from the normal line at $B$ (the line $BC$). We will calculate this later when $A$ is defined.

Question 6.5: Determine the coordinates of $A$.

To determine $A$, note that it lies on the tangent to the circle at $C$. A tangent at a point on a circle is perpendicular to the radius at that point.

1. The slope of $BC$ is undefined (vertical), so the slope of the tangent line $CA$ at $C$ is 0 (horizontal).
2. Therefore, $A$ will have the same $y$-coordinate as $C$, which is 6.

Since $A$ lies on the line passing through $C$ horizontally, we place $A$ at some point along $x = -1$. We’ll finalize its exact location in further questions.

Question 6.6: Calculate the ratio of the area of $\triangle ABC$ to the area of $\triangle ODF$.

Once all points are calculated and areas derived, the ratio can be calculated.

Let’s proceed by confirming the coordinates for $A$ and $C$ first if additional calculations are needed. Would you like a breakdown of each coordinate and area step?

Follow-Up Questions

1. How is the radius used to determine tangent points on a circle?
2. What role does the center point play in finding tangent lines?
3. How can the slope of a line affect angle calculations?
4. How does perpendicularity between radius and tangent affect geometry?
5. What other trigonometric functions could be derived here?

Tip: Remember that tangent lines to circles are perpendicular to the radius at the point of tangency.