ENDPOINTS OF DIAMETER OF A CIRCLE

Solved Problems

Problem 1 :

If the two endpoints of diameter are (2, 3) and (-1, 5), find the center of the circle.

Solution :

Center of the circle is the midpoint of the two end points of diameter.

Midpoint of the line segment joining (x₁, y₁) and (x₂, y₂).

Substitute (x₁, y₁) = (2, 3) and (x₂, y₂) = (-1, 5).

= (½, 4)

Therefore, the center of the circle is (½, 4).

Problem 2 :

If the two endpoints of diameter are (, 4) and (-1, -2), find the length of the diameter.

Solution :

Lenth of diameter in a circle is the distance between the two endpoints of the diameter.

Distance between the two points (x₁, y₁) and (x₂, y₂) :

Substitute (x₁, y₁) = (8, 7) and (x₂, y₂) = (2, 10).

= 3√5

Therefore, the length of the diameter is 3√5 units.

Problem 3 :

If the two endpoints of diameter are (7, 4) and (-1, -2), find the radius of the circle.

Solution :

Distance between the two points (x₁, y₁) and (x₂, y₂) :

Substitute (x₁, y₁) = (7, 4) and (x₂, y₂) = (-1, -2) to find the length of diameter.

= √100

= 10 units

= ¹⁰⁄₂

= 5 units

Problem 4 :

The center of a circle is (-2, 3). If one end endpoint of diameter is (3, -5), find the other end point.

Solution :

We already know that the midpoint of two end points of diameter is equal to center of the circle.

Let (x₁, y₁) and (x₂, y₂) be the two end points of the diameter. Then, we have

One endpoint of the diameter is (-2, 3). Let (a, b) be the pother end point of the diameter.

Substitute (x₁, y₁) = (-2, 3) and (x₂, y₂) = (a, b).

Equate the x-coordinates and y-coordinates.

(a, b) = (-2, 3)

The other end point of the diameter is (-2, 3).

Problem 5 :

If the two endpoints of diameter of a circle are (2, 3) and (6, -5), find the equation of the cricle in standard form.

Solution :

Center of the circle :

= midpoint of the end points of diameter

= (4, -1)

Distance between one end point of diameter (2, 3) and center of the circle (4, 1).

Radius = √8 units

Equation of a circle in standard form :

(x - h)² + (y - k)² = r²

where, center = (h, k) and radius = r.

Substitute (h, k) = (4, 1) and r = √8.

(x - 4)² + (y - 1)² = (√8)²

(x - 4)² + (y - 1)² = 8

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