The given problem describes a circle $C$ with equation:

$(x + 5)^2 + (y + 3)^2 = 80$

This is the standard form of a circle equation, where the center of the circle is at $(-5, -3)$, and the radius $r$ is:

$r = \sqrt{80} = 4\sqrt{5}$

The line $l$ is a tangent to the circle, with the gradient $m = 2$. We are asked to find two possible equations for $l$ in the form $y = mx + c$.

Step-by-Step Solution:

1. Equation of the Tangent Line: The general equation of a line is $y = mx + c$, where $m$ is the gradient (given as 2), and $c$ is the y-intercept. Therefore, we know the general form for our tangent lines will be:

   $y = 2x + c$

2. Condition for Tangency: A line is tangent to a circle if the perpendicular distance from the center of the circle to the line is equal to the radius of the circle.

   The formula for the perpendicular distance $d$ from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$ is:

   $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

   In this case, the line is $2x - y + c = 0$ (rearranged form), and the center of the circle is $(-5, -3)$. Substituting these into the distance formula:

   $d = \frac{|2(-5) - (-3) + c|}{\sqrt{2^2 + (-1)^2}} = \frac{| -10 + 3 + c |}{\sqrt{5}} = \frac{| -7 + c |}{\sqrt{5}}$

3. Equating Distance to the Radius: Since the line is a tangent, the distance $d$ must be equal to the radius of the circle, which is $4\sqrt{5}$:

   $\frac{| -7 + c |}{\sqrt{5}} = 4\sqrt{5}$

   Multiply both sides by $\sqrt{5}$:

   $| -7 + c | = 20$

4. Solve for $c$: The absolute value equation gives us two cases:

   • Case 1: $-7 + c = 20$, which gives $c = 27$
   • Case 2: $-7 + c = -20$, which gives $c = -13$

5. Equations of the Tangent Lines: Therefore, the two possible equations for the tangent lines are:

   $y = 2x + 27$ $y = 2x - 13$

These are the two required equations for the tangent lines to the circle.

Would you like more details or have any questions?

Follow-up Questions:

1. How do you find the center and radius of a circle from its equation?
2. What is the condition for a line to be tangent to a circle?
3. How do you use the distance formula for a point and a line?
4. What are the steps to solve an absolute value equation?
5. How can you verify that a line is tangent to a circle?

Tip:

For a quick check if a line is tangent to a circle, compute the discriminant of the quadratic equation formed by substituting the line into the circle equation. If it is zero, the line is tangent!