Question

try: find the equation of the circles with given centres and radii. (i) centre: (5,4) and radius, r = 6 (i) centre: (-2,4) and radius, r = 6 (iii) centre: (-1, -3) and radius, r = 5 (iv) centre: (-3, -2) and radius, r = 5

Explanation:

To find the equation of a circle with center \((h, k)\) and radius \(r\), we use the standard form of the circle's equation: \(\boldsymbol{(x - h)^2 + (y - k)^2 = r^2}\). We will solve each part one by one.

(i) Centre: \((5, 4)\) and radius, \(r = 6\)

Step 1: Identify \(h\), \(k\), and \(r\)

Here, \(h = 5\), \(k = 4\), and \(r = 6\).

Step 2: Substitute into the circle's equation

Substitute \(h = 5\), \(k = 4\), and \(r = 6\) into \((x - h)^2 + (y - k)^2 = r^2\):
\[
(x - 5)^2 + (y - 4)^2 = 6^2
\]

Step 3: Simplify \(r^2\)

Since \(6^2 = 36\), the equation becomes:
\[
(x - 5)^2 + (y - 4)^2 = 36
\]

(ii) Centre: \((-2, 4)\) and radius, \(r = 6\)

Step 1: Identify \(h\), \(k\), and \(r\)

Here, \(h = -2\), \(k = 4\), and \(r = 6\).

Step 2: Substitute into the circle's equation

Substitute \(h = -2\), \(k = 4\), and \(r = 6\) into \((x - h)^2 + (y - k)^2 = r^2\). Note that \((x - (-2)) = (x + 2)\):
\[
(x + 2)^2 + (y - 4)^2 = 6^2
\]

Step 3: Simplify \(r^2\)

Since \(6^2 = 36\), the equation becomes:
\[
(x + 2)^2 + (y - 4)^2 = 36
\]

(iii) Centre: \((-1, -3)\) and radius, \(r = 5\)

Step 1: Identify \(h\), \(k\), and \(r\)

Here, \(h = -1\), \(k = -3\), and \(r = 5\).

Step 2: Substitute into the circle's equation

Substitute \(h = -1\), \(k = -3\), and \(r = 5\) into \((x - h)^2 + (y - k)^2 = r^2\). Note that \((x - (-1)) = (x + 1)\) and \((y - (-3)) = (y + 3)\):
\[
(x + 1)^2 + (y + 3)^2 = 5^2
\]

Step 3: Simplify \(r^2\)

Since \(5^2 = 25\), the equation becomes:
\[
(x + 1)^2 + (y + 3)^2 = 25
\]

(iv) Centre: \((-3, -2)\) and radius, \(r = 5\)

Step 1: Identify \(h\), \(k\), and \(r\)

Here, \(h = -3\), \(k = -2\), and \(r = 5\).

Step 2: Substitute into the circle's equation

Substitute \(h = -3\), \(k = -2\), and \(r = 5\) into \((x - h)^2 + (y - k)^2 = r^2\). Note that \((x - (-3)) = (x + 3)\) and \((y - (-2)) = (y + 2)\):
\[
(x + 3)^2 + (y + 2)^2 = 5^2
\]

Step 3: Simplify \(r^2\)

Since \(5^2 = 25\), the equation becomes:
\[
(x + 3)^2 + (y + 2)^2 = 25
\]

Final Answers:

• (i) \(\boldsymbol{(x - 5)^2 + (y - 4)^2 = 36}\)
• (ii) \(\boldsymbol{(x + 2)^2 + (y - 4)^2 = 36}\)
• (iii) \(\boldsymbol{(x + 1)^2 + (y + 3)^2 = 25}\)
• (iv) \(\boldsymbol{(x + 3)^2 + (y + 2)^2 = 25}\)

Answer:

To find the equation of a circle with center \((h, k)\) and radius \(r\), we use the standard form of the circle's equation: \(\boldsymbol{(x - h)^2 + (y - k)^2 = r^2}\). We will solve each part one by one.

(i) Centre: \((5, 4)\) and radius, \(r = 6\)

Step 1: Identify \(h\), \(k\), and \(r\)

Here, \(h = 5\), \(k = 4\), and \(r = 6\).

Step 2: Substitute into the circle's equation

Substitute \(h = 5\), \(k = 4\), and \(r = 6\) into \((x - h)^2 + (y - k)^2 = r^2\):
\[
(x - 5)^2 + (y - 4)^2 = 6^2
\]

Step 3: Simplify \(r^2\)

Since \(6^2 = 36\), the equation becomes:
\[
(x - 5)^2 + (y - 4)^2 = 36
\]

(ii) Centre: \((-2, 4)\) and radius, \(r = 6\)

Step 1: Identify \(h\), \(k\), and \(r\)

Here, \(h = -2\), \(k = 4\), and \(r = 6\).

Step 2: Substitute into the circle's equation

Substitute \(h = -2\), \(k = 4\), and \(r = 6\) into \((x - h)^2 + (y - k)^2 = r^2\). Note that \((x - (-2)) = (x + 2)\):
\[
(x + 2)^2 + (y - 4)^2 = 6^2
\]

Step 3: Simplify \(r^2\)

Since \(6^2 = 36\), the equation becomes:
\[
(x + 2)^2 + (y - 4)^2 = 36
\]

(iii) Centre: \((-1, -3)\) and radius, \(r = 5\)

Step 1: Identify \(h\), \(k\), and \(r\)

Here, \(h = -1\), \(k = -3\), and \(r = 5\).

Step 2: Substitute into the circle's equation

Substitute \(h = -1\), \(k = -3\), and \(r = 5\) into \((x - h)^2 + (y - k)^2 = r^2\). Note that \((x - (-1)) = (x + 1)\) and \((y - (-3)) = (y + 3)\):
\[
(x + 1)^2 + (y + 3)^2 = 5^2
\]

Step 3: Simplify \(r^2\)

Since \(5^2 = 25\), the equation becomes:
\[
(x + 1)^2 + (y + 3)^2 = 25
\]

(iv) Centre: \((-3, -2)\) and radius, \(r = 5\)

Step 1: Identify \(h\), \(k\), and \(r\)

Here, \(h = -3\), \(k = -2\), and \(r = 5\).

Step 2: Substitute into the circle's equation

Substitute \(h = -3\), \(k = -2\), and \(r = 5\) into \((x - h)^2 + (y - k)^2 = r^2\). Note that \((x - (-3)) = (x + 3)\) and \((y - (-2)) = (y + 2)\):
\[
(x + 3)^2 + (y + 2)^2 = 5^2
\]

Step 3: Simplify \(r^2\)

Since \(5^2 = 25\), the equation becomes:
\[
(x + 3)^2 + (y + 2)^2 = 25
\]

Final Answers:

• (i) \(\boldsymbol{(x - 5)^2 + (y - 4)^2 = 36}\)
• (ii) \(\boldsymbol{(x + 2)^2 + (y - 4)^2 = 36}\)
• (iii) \(\boldsymbol{(x + 1)^2 + (y + 3)^2 = 25}\)
• (iv) \(\boldsymbol{(x + 3)^2 + (y + 2)^2 = 25}\)