The image contains exercises involving geometry and sequences. Here’s the breakdown of what I see in the tasks and how to approach them:

EXERCICE 1

This is a geometry exercise dealing with points in a plane and circles.

1. Description of points $A(4,3)$ and $B(8,6)$:

   • The exercise asks to analyze these points and derive properties in the plane.

2. Task a:

   • Prove that $C$, defined as $(x,y)$ satisfying $AM = AN$, lies on a specific circle.

3. Task b:

   • Show that $C$ is on a circle with center $H(2,4)$ and radius $R = \sqrt{5}$.

4. Task c:

   • Derive the equation of a line $D$ tangent to the circle $C$ at $H$.

5. Task d:

   • Find the coordinates of two points where line $D$ intersects the circle.

EXERCICE 2

This exercise deals with numerical sequences and their properties.

1. Task a:

   • Demonstrate the given sequence $U_n$ by induction.

2. Task b:

   • Verify the recurrence relation $U_{n+1} = \frac{U_n}{U_n - 2}$.

3. Task c:

   • Study the monotonicity of $U_n$.

4. Task d:

   • Deduce properties about the sequence.

Suggestions for Solving

For clarity, I’ll briefly outline solutions and methods for some of these tasks.

EXERCICE 1

• (a) To show $C$ satisfies the circle equation:

  • Use the condition $AM = AN$, which represents the locus of all points equidistant from $A$ and $B$, forming the perpendicular bisector of $AB$. This bisector is the circle.

• (b) Confirm that $H(2,4)$ is the center and $R = \sqrt{5}$ by computing the distance using the equation derived in (a).

• (c) The tangent line equation can be determined using the slope perpendicular to the radius at point $H$.

• (d) Solve for intersection points of the tangent and circle equations.

EXERCICE 2

• (a) Use mathematical induction to verify the formula for $U_n$.

• (b) Substitute $U_{n+1} = \frac{U_n}{U_n - 2}$ into the recurrence relation and simplify to confirm consistency.

• (c) Study monotonicity by analyzing the derivative or comparing successive terms $U_n$ and $U_{n+1}$.

Would you like me to solve a specific part in detail or provide further insights into the problems? Let me know!

Follow-up Questions:

1. How do you compute the equation of a circle given its center and radius?
2. What is the process for deriving the tangent line to a circle at a specific point?
3. How do you confirm monotonicity in a sequence rigorously?
4. What are the steps to solve a system of equations involving line-circle intersections?
5. How does mathematical induction work for verifying sequence properties?

Tip:

For geometry problems, always start by visualizing the elements (points, lines, circles) to gain a better understanding of their relationships.