Question

section b: analytic geometry

1. a circle with centre at (0,0) passes through the point (-12,5).

a) find the radius of this circle.

b) state the equation of this circle.

c) sketch the graph. label clearly the intercepts on the graph

d) state any 2 other points that lie on the circle.

Explanation:

Find the radius

We use the distance formula from the center \((0,0)\) to the point \((-12,5)\):

$$ r = \sqrt{(-12 - 0)^2 + (5 - 0)^2} $$

$$ r = \sqrt{144 + 25} = \sqrt{169} = 13 $$

State the equation

The standard equation of a circle centered at the origin \((0,0)\) is:

$$ x^2 + y^2 = r^2 $$

Substituting \(r = 13\):

$$ x^2 + y^2 = 13^2 \implies x^2 + y^2 = 169 $$

Identify the intercepts

The intercepts of the circle with radius \(13\) centered at the origin are:

• \(x\)-intercepts: \((13, 0)\) and \((-13, 0)\)
• \(y\)-intercepts: \((0, 13)\) and \((0, -13)\)

Find other points

Using the symmetry of the circle and the given point \((-12,5)\), we can find other integer coordinate points:

• Reflecting across the \(y\)-axis: \((12, 5)\)
• Reflecting across the \(x\)-axis: \((-12, -5)\)
• Reflecting both: \((12, -5)\)
• Swapping coordinates: \((5, 12)\), \((-5, 12)\), \((5, -12)\), \((-5, -12)\)

Answer:

Question 1

a) The radius of the circle is \(13\).

b) The equation of the circle is:

$$ x^2 + y^2 = 169 $$

c) The graph is a circle centered at \((0,0)\) with a radius of \(13\). The intercepts are:

• \(x\)-intercepts: \((13, 0)\) and \((-13, 0)\)
• \(y\)-intercepts: \((0, 13)\) and \((0, -13)\)

d) Two other points that lie on the circle are \((12, 5)\) and \((12, -5)\).