B2. Dani sta enačbi $3 x-2=-x+10$ in $\frac{y}{2}-\frac{y}{3}=\frac{1}{3}$.

(a) Rešite enačbi.

(b) Rešitvi $x$ in $y$ sta koordinati točke $S$, ki predstavlja središče kroga, ki se dotika $y$ osi. Narišite in označite središče ter narišite krožnico v koordinatnem sistemu.

(c) Poimenujte del kroga, ki se nahaja pod $x$ osjo.

(d) Narišite tangento, ki je vzporedna $x$ osi in zapišite njeno enačbo.

(e) Izračunajte koordinati enega izmed presečišč krožnice $\mathrm{z} x$ osjo na dve decimalni mesti natančno.

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B2. The equations given are $3x - 2 = -x + 10$ and $\frac{y}{2} - \frac{y}{3} = \frac{1}{3}$.

(a) Solve the equations.

(b) The solutions $x$ and $y$ are the coordinates of point $S$, which represents the center of a circle that touches the $y$-axis. Draw and label the center, and draw the circle in the coordinate system.

(c) Name the part of the circle that lies below the $x$-axis.

(d) Draw the tangent that is parallel to the $x$-axis and write its equation.

(e) Calculate the coordinates of one of the intersections of the circle with the $x$-axis to two decimal places.
B2. The equation $3 x-2=-x+10$ has the solution $x=3$, and the equation $\frac{y}{2}-\frac{y}{3}=\frac{1}{3}$ has the solution $y=2$.

Below the $y$-axis, there is a circular segment.

The $x$-axis is parallel to two tangents: $y=-1$ and $y=5$.

A right triangle is formed between the center of the circle, the intersection of the circle with the $x$-axis, and the point $(3,0)$. For this triangle, we can write the Pythagorean theorem as $x^{2}+2^{2}=3^{2}$, where $x$ is the distance between the intersection of the circle with the $x$-axis and the point $(3,0)$. Solving the equation, we get the solution $x=2.24$. Considering this distance $x=2.24$, we determine the coordinates of the intersections of the circle with the $x$-axis: $x_{1}=3-2.24=0.76$ and

$x_{2}=3+2.24=5.24$.

![](https://cdn.mathpix.com/cropped/2024_06_07_b93fa657b92bc897cb7bg-11.jpg?height=960&width=1130&top_left_y=845&top_left_x=560)

(a) The calculated solutions of the system $x=3$ and $y=2$

(b) Correctly drawn point $\mathrm{S}$ and the circle that touches the $y$-axis (the contestant also gets a point if the coordinates are incorrectly calculated)

(c) The shape below the $x$-axis is named: circular segment.

$1 \mathrm{t}$

(d) Writing the equation and graphically representing at least one of the tangents $y=-1$ or $y=5$

(e) Applying the Pythagorean theorem to calculate the distance $x=2.24 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ t

Calculating one of the coordinates of the intersection of the circle with the $x$-axis: $x_{1}=0.76$ or $x_{2}=5.24$. . .... $1 \mathrm{t}$