Question

practice: find the equation of the circles with the given pairs of coordinates at the endpoints of diameter: (i) (4,5) and (-3,-4) (ii) (2, -5) and (-1, -6) (iii) (3,4) and (8,5)

Explanation:

To find the equation of a circle when the endpoints of its diameter are given, we use the diameter form of the circle's equation:

If \((x_1, y_1)\) and \((x_2, y_2)\) are the endpoints of the diameter, the equation of the circle is:
\[
(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0
\]

Part (i): Endpoints \((4, 5)\) and \((-3, -4)\)

Let \((x_1, y_1) = (4, 5)\) and \((x_2, y_2) = (-3, -4)\).

Step 1: Substitute into the diameter form

\[
(x - 4)(x - (-3)) + (y - 5)(y - (-4)) = 0
\]
Simplify the signs:
\[
(x - 4)(x + 3) + (y - 5)(y + 4) = 0
\]

Step 2: Expand the products

• Expand \((x - 4)(x + 3)\):

\[
x^2 + 3x - 4x - 12 = x^2 - x - 12
\]

• Expand \((y - 5)(y + 4)\):

\[
y^2 + 4y - 5y - 20 = y^2 - y - 20
\]

Step 3: Combine terms

Add the expanded forms and set to 0:
\[
x^2 - x - 12 + y^2 - y - 20 = 0
\]
Simplify:
\[
x^2 + y^2 - x - y - 32 = 0
\]

Part (ii): Endpoints \((2, -5)\) and \((-1, -6)\)

Let \((x_1, y_1) = (2, -5)\) and \((x_2, y_2) = (-1, -6)\).

Step 1: Substitute into the diameter form

\[
(x - 2)(x - (-1)) + (y - (-5))(y - (-6)) = 0
\]
Simplify the signs:
\[
(x - 2)(x + 1) + (y + 5)(y + 6) = 0
\]

Step 2: Expand the products

• Expand \((x - 2)(x + 1)\):

\[
x^2 + x - 2x - 2 = x^2 - x - 2
\]

• Expand \((y + 5)(y + 6)\):

\[
y^2 + 6y + 5y + 30 = y^2 + 11y + 30
\]

Step 3: Combine terms

Add the expanded forms and set to 0:
\[
x^2 - x - 2 + y^2 + 11y + 30 = 0
\]
Simplify:
\[
x^2 + y^2 - x + 11y + 28 = 0
\]

Part (iii): Endpoints \((3, 4)\) and \((8, 5)\)

Let \((x_1, y_1) = (3, 4)\) and \((x_2, y_2) = (8, 5)\).

Step 1: Substitute into the diameter form

\[
(x - 3)(x - 8) + (y - 4)(y - 5) = 0
\]

Step 2: Expand the products

• Expand \((x - 3)(x - 8)\):

\[
x^2 - 8x - 3x + 24 = x^2 - 11x + 24
\]

• Expand \((y - 4)(y - 5)\):

\[
y^2 - 5y - 4y + 20 = y^2 - 9y + 20
\]

Step 3: Combine terms

Add the expanded forms and set to 0:
\[
x^2 - 11x + 24 + y^2 - 9y + 20 = 0
\]
Simplify:
\[
x^2 + y^2 - 11x - 9y + 44 = 0
\]

Final Answers

(i) \(\boldsymbol{x^2 + y^2 - x - y - 32 = 0}\)
(ii) \(\boldsymbol{x^2 + y^2 - x + 11y + 28 = 0}\)
(iii) \(\boldsymbol{x^2 + y^2 - 11x - 9y + 44 = 0}\)

Answer:

To find the equation of a circle when the endpoints of its diameter are given, we use the diameter form of the circle's equation:

If \((x_1, y_1)\) and \((x_2, y_2)\) are the endpoints of the diameter, the equation of the circle is:
\[
(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0
\]

Part (i): Endpoints \((4, 5)\) and \((-3, -4)\)

Let \((x_1, y_1) = (4, 5)\) and \((x_2, y_2) = (-3, -4)\).

Step 1: Substitute into the diameter form

\[
(x - 4)(x - (-3)) + (y - 5)(y - (-4)) = 0
\]
Simplify the signs:
\[
(x - 4)(x + 3) + (y - 5)(y + 4) = 0
\]

Step 2: Expand the products

• Expand \((x - 4)(x + 3)\):

\[
x^2 + 3x - 4x - 12 = x^2 - x - 12
\]

• Expand \((y - 5)(y + 4)\):

\[
y^2 + 4y - 5y - 20 = y^2 - y - 20
\]

Step 3: Combine terms

Add the expanded forms and set to 0:
\[
x^2 - x - 12 + y^2 - y - 20 = 0
\]
Simplify:
\[
x^2 + y^2 - x - y - 32 = 0
\]

Part (ii): Endpoints \((2, -5)\) and \((-1, -6)\)

Let \((x_1, y_1) = (2, -5)\) and \((x_2, y_2) = (-1, -6)\).

Step 1: Substitute into the diameter form

\[
(x - 2)(x - (-1)) + (y - (-5))(y - (-6)) = 0
\]
Simplify the signs:
\[
(x - 2)(x + 1) + (y + 5)(y + 6) = 0
\]

Step 2: Expand the products

• Expand \((x - 2)(x + 1)\):

\[
x^2 + x - 2x - 2 = x^2 - x - 2
\]

• Expand \((y + 5)(y + 6)\):

\[
y^2 + 6y + 5y + 30 = y^2 + 11y + 30
\]

Step 3: Combine terms

Add the expanded forms and set to 0:
\[
x^2 - x - 2 + y^2 + 11y + 30 = 0
\]
Simplify:
\[
x^2 + y^2 - x + 11y + 28 = 0
\]

Part (iii): Endpoints \((3, 4)\) and \((8, 5)\)

Let \((x_1, y_1) = (3, 4)\) and \((x_2, y_2) = (8, 5)\).

Step 1: Substitute into the diameter form

\[
(x - 3)(x - 8) + (y - 4)(y - 5) = 0
\]

Step 2: Expand the products

• Expand \((x - 3)(x - 8)\):

\[
x^2 - 8x - 3x + 24 = x^2 - 11x + 24
\]

• Expand \((y - 4)(y - 5)\):

\[
y^2 - 5y - 4y + 20 = y^2 - 9y + 20
\]

Step 3: Combine terms

Add the expanded forms and set to 0:
\[
x^2 - 11x + 24 + y^2 - 9y + 20 = 0
\]
Simplify:
\[
x^2 + y^2 - 11x - 9y + 44 = 0
\]

Final Answers

(i) \(\boldsymbol{x^2 + y^2 - x - y - 32 = 0}\)
(ii) \(\boldsymbol{x^2 + y^2 - x + 11y + 28 = 0}\)
(iii) \(\boldsymbol{x^2 + y^2 - 11x - 9y + 44 = 0}\)