Analytical geometry examples

Analytical geometry solved examples


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Circle given by two points and tangent to the y axis

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Given two points on a circle (6 , 2) and (3 , -1), the circle is tangent to the y axis. Find the radius and the center coordinate of the circle.

Draw the circle and mark the point of the circle center that is at a = R and b (because the circle is tangent to the y axis, b is unknown)

We can write the equation of the circle related to the points and the center of the circle:

( x₁ − R )² + ( y₁ − b )² = R²	(1)
( x₂ − R )² + ( y₂ − b )² = R²	(2)

From first eq. (1)	x₁² − 2x₁R + R² + y₁² − 2y₁b + b² = R²
	x₁² − 2x₁R + y₁² − 2y₁b + b² = 0

From first eq. (2)	x₂² − 2x₂R + R² + y₂² − 2y₂b + b² = R²
	x₂² − 2x₂R + y₂² − 2y₂b + b² = 0

Substitute R from eq. (3) into the last equation and arrange terms:

b²(2x₁ − 2x₂) + b(4x₂y₁ − 4x₁y₂) + 2x₁x₂² − 2x₁²x₂ + 2x₁y₂² − 2x₂y₁² = 0

b²(x₁ − x₂) + b(2x₂y₁ − 2x₁y₂) + x₁x₂² − x₁²x₂ + x₁y₂² − x₂y₁² = 0

We get a quadratic equation with the unknown b (all the points are given) the solution is:

Now insert the value of b into eq. (3) to get the radius R of the circle.

Insert the given values of the points (6 , 2) and (3 , − 1) to get b:

The radius from eq. (3) for b = 2 is:
And the second radius for b = − 10 is:

We can see that there are two circles that fulfills the requirements of the given data.

The two circles centers are at the points (3 , 2) and (15 , − 10)

Circle given by two points and the radius

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Given two points on a circle (2 , -1) and (-2 , 7) and the radius which is equal to 5. Find the center coordinate of the circle.

We can write the equation of the circle with two points which are located on the circle as:

(x₁ - a)² + (y₁ - b)² = R²

(x₂ - a)² + (y₂ - b)² = R²

We get two equations with two unknowns a and b but to find their values from this equations is not an easy job, you can try it.
So we will try another method to solve the problem.

Connect the two points so it make a line
The mid point coordinate of this two points is:

x_m = (x₁ + x₂) / 2

y_m = (y₁ + y₂) / 2

The slope of the line connecting the two points is:
Line AB is perpendicular to this line and the slope is:

The equation of line AB based on point (x_m , y_m) is:

y − y_m = m_AB (x − x_m)

Line AB is passing through the center of the circle (point a,b) so now we search a point on this line which is at a distance of R fron the points (x₁ , y₁), so we have the equations