Let $a,b\in \mathbb{R}$ such that $b>a^2$. Find all the matrices $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A^2-2aA+bI_2)=0$.
Given \( a, b \in \mathbb{R} \) such that \( b > a^2 \), we need to find all matrices \( A \in \mathcal{M}_2(\mathbb{R}) \) such that \( \det(A^2 - 2aA + bI_2) = 0 \).

1. Let \( \lambda_1 \) and \( \lambda_2 \) be the eigenvalues of \( A \). Then, the characteristic polynomial of \( A \) is given by:
   \[
   \det(A - \lambda I) = (\lambda_1 - \lambda)(\lambda_2 - \lambda)
   \]
   Since \( \det(A^2 - 2aA + bI_2) = 0 \), it follows that:
   \[
   (\lambda_1^2 - 2a \lambda_1 + b)(\lambda_2^2 - 2a \lambda_2 + b) = 0
   \]
   This implies that at least one of the factors must be zero:
   \[
   \lambda_1^2 - 2a \lambda_1 + b = 0 \quad \text{or} \quad \lambda_2^2 - 2a \lambda_2 + b = 0
   \]

2. Solving the quadratic equation \( \lambda^2 - 2a \lambda + b = 0 \), we get:
   \[
   \lambda = a \pm \sqrt{a^2 - b}
   \]
   Given \( b > a^2 \), the term under the square root is negative, so the eigenvalues are complex:
   \[
   \lambda_1 = a + i \sqrt{b - a^2}, \quad \lambda_2 = a - i \sqrt{b - a^2}
   \]

3. The trace and determinant of \( A \) are related to its eigenvalues by:
   \[
   \text{tr}(A) = \lambda_1 + \lambda_2 = 2a
   \]
   \[
   \det(A) = \lambda_1 \lambda_2 = (a + i \sqrt{b - a^2})(a - i \sqrt{b - a^2}) = a^2 + (b - a^2) = b
   \]

4. By the Cayley-Hamilton theorem, any matrix \( A \) satisfies its own characteristic polynomial. For a \( 2 \times 2 \) matrix \( A \) with eigenvalues \( \lambda_1 \) and \( \lambda_2 \), the characteristic polynomial is:
   \[
   p(\lambda) = \lambda^2 - (\lambda_1 + \lambda_2)\lambda + \lambda_1 \lambda_2 = \lambda^2 - 2a\lambda + b
   \]
   Therefore, \( A \) satisfies:
   \[
   A^2 - 2aA + bI = 0
   \]
   This implies:
   \[
   \det(A^2 - 2aA + bI) = 0
   \]

5. Hence, the matrices \( A \in \mathcal{M}_2(\mathbb{R}) \) such that \( \det(A^2 - 2aA + bI) = 0 \) are those for which \( \text{tr}(A) = 2a \) and \( \det(A) = b \).

The final answer is \( \boxed{\text{Matrices } A \in \mathcal{M}_2(\mathbb{R}) \text{ such that } \text{tr}(A) = 2a \text{ and } \det(A) = b} \).