Find the distance of the centre of the circle x^2+y^2+z^2+x-2y+2z=3,2x+y+2z=1 from the plane ax+by+cz=d, where a,b,c,d are constants.?
See belo.
Given the Sphere
and the plane
with
The circle
but
now given
with
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To find the distance between the center of the circle and the plane, we need to follow these steps:
- Rewrite the equations of the circle and the plane in standard form.
- Find the center of the circle.
- Use the formula for the distance between a point and a plane to find the distance.
Given equations: Circle: (x^2 + y^2 + z^2 + x - 2y + 2z = 3) Plane: (ax + by + cz = d)
Step 1: Rewrite equations in standard form: Circle: (x^2 + y^2 + z^2 + x - 2y + 2z - 3 = 0) Plane: (ax + by + cz - d = 0)
Step 2: Complete the square for the circle equation to find the center. ((x^2 + x) + (y^2 - 2y) + (z^2 + 2z) = 3) ((x + \frac{1}{2})^2 - \frac{1}{4} + (y - 1)^2 - 1 + (z + 1)^2 - 1 = 3) ((x + \frac{1}{2})^2 + (y - 1)^2 + (z + 1)^2 = \frac{15}{4}) Center of the circle: ((- \frac{1}{2}, 1, -1))
Step 3: Use the distance formula for a point and a plane. Distance (= \frac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}})
Substitute the values: Distance (= \frac{|a(-\frac{1}{2}) + b(1) + c(-1) - d|}{\sqrt{a^2 + b^2 + c^2}})
Thus, the distance between the center of the circle and the plane is given by the above expression.
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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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