the curves $$ x^{2}+y^{2}=10 $$ and $$ x^{2}+y^{2}+a y=b $$ intersect at right angles where $$ x=3 $$ ind the possible values of $$ a $$ and $$ b $$.

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To determine the possible values of $$ a $$ and $$ b $$ such that the curves $$ x^{2} + y^{2} = 10 $$ and $$ x^{2} + y^{2} + a y = b $$ intersect at right angles where $$ x = 3 $$, follow these steps:

### 1. Identify the Points of Intersection
When $$ x = 3 $$, substitute into the first equation:
$$
3^{2} + y^{2} = 10 \implies 9 + y^{2} = 10 \implies y^{2} = 1 \implies y = \pm 1
$$
Thus, the points of intersection are $$ (3, 1) $$ and $$ (3, -1) $$.

### 2. Condition for Orthogonal Intersection
For two curves to intersect orthogonally (at right angles), the gradients (derivatives) of their tangent lines at the point of intersection must satisfy:
$$
\nabla F \cdot \nabla G = 0
$$
where $$ F(x, y) = x^2 + y^2 - 10 $$ and $$ G(x, y) = x^2 + y^2 + a y - b $$.

Compute the gradients:
$$
\nabla F = (2x, 2y) \
\nabla G = (2x, 2y + a)
$$
Their dot product is:
$$
2x \cdot 2x + 2y \cdot (2y + a) = 4x^2 + 4y^2 + 2a y = 0
$$
Using the first curve's equation $$ x^2 + y^2 = 10 $$:
$$
4(10) + 2a y = 0 \implies 40 + 2a y = 0 \implies a y = -20
$$

### 3. Determine Possible Values of $$ a $$ and $$ b $$
At the intersection points $$ (3, 1) $$ and $$ (3, -1) $$:

- For $$ (3, 1) $$:
  $$
  a \cdot 1 = -20 \implies a = -20
  $$
  Substitute into the second equation:
  $$
  9 + 1 - 20 \cdot 1 = b \implies b = -10
  $$

- For $$ (3, -1) $$:
  $$
  a \cdot (-1) = -20 \implies a = 20
  $$
  Substitute into the second equation:
  $$
  9 + 1 + 20 \cdot (-1) = b \implies b = -10
  $$

### 4. Conclusion
There are two possible pairs of $$ (a, b) $$ that satisfy the conditions:

$$
(a, b) = (-20, -10) \quad \text{and} \quad (a, b) = (20, -10)
$$

**Answer:** All real numbers a and b with b = –10 and a equal to –20 or 20. Thus,

a = −20 and b = −10 or a = 20 and b = −10
The possible values are $$ a = -20 $$ and $$ b = -10 $$, or $$ a = 20 $$ and $$ b = -10 $$.

the curves and intersect at right angles where
ind the possible values of and .

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the curves and intersect at right angles where ind the possible values of and .