How do you find the equation of the circle with center at (1, 3) and tangent to the line whose equation is x – y + 2 =0?

Answer 1

There is no such circle.

The line #x-y+2=0# contains the point #(1,3)#.

A line through the center of a circle is not tangent to the circle.

Therefore, there is no circle matching the description.

Suggestion

I saw the answer when I tried to draw a sketch of the situation. The line has slope-intercept equation #y=x+2#. Whe I tried to graph the line starting ay the #y# intercept and using the slope, it ran through the point #(1,3)#.
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Answer 2

To find the equation of the circle with center at (1, 3) and tangent to the line x – y + 2 = 0, we can use the distance formula between a point and a line.

First, we need to find the distance between the center of the circle and the line. The distance formula is given by:

Distance = |Ax + By + C| / √(A^2 + B^2)

In this case, the line equation is x – y + 2 = 0, which can be rewritten as x - y = -2. Comparing this to the standard form Ax + By + C = 0, we have A = 1, B = -1, and C = -2.

Substituting these values into the distance formula, we get:

Distance = |1(1) + (-1)(3) + (-2)| / √(1^2 + (-1)^2) = |1 - 3 - 2| / √(1 + 1) = |-4| / √2 = 4 / √2 = 2√2

Since the circle is tangent to the line, the distance between the center of the circle and the line is equal to the radius of the circle. Therefore, the radius of the circle is 2√2.

The equation of a circle with center (h, k) and radius r is given by:

(x - h)^2 + (y - k)^2 = r^2

Substituting the given values, we have:

(x - 1)^2 + (y - 3)^2 = (2√2)^2 (x - 1)^2 + (y - 3)^2 = 8

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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