The equation of a circle can be found using the centre and radius. The discriminant can determine the nature of intersections between two circles or a circle and a line to prove for tangency.

What is the distance between the points \((3,4)\) and \(( - 1, - 2)\)?

\[\sqrt {20}\]

\[\sqrt {40}\]

\[\sqrt {52}\]

What is the distance between the points \(( - 1,6)\) and \((2, - 4)\)?

\[\sqrt {109}\]

\[\sqrt {89}\]

\[\sqrt {66}\]

If the point \(( - 2,p)\) lies on the circle with equation \({x^2} + {y^2} = 36\) then what is \(p\) equal to?

\[\pm \sqrt {40}\]

\[\pm \sqrt {32}\]

\[\pm \sqrt {38}\]

Do the pair of circles touch, intersect or miss?

\({(x + 1)^2} + {(y + 4)^2} = 13\) and \({(x - 7)^2} + {(y - 13)^2} = 52\)

The circles touch externally

The circles intersect

The circles miss

\({(x - 1)^2} + {(y - 5)^2} = 13\) and \({(x - 7)^2} + {(y - 14)^2} = 52\)

The circles touch internally

\({(x - 8)^2} + {(y - 14)^2} = 13\) and \({(x - 7)^2} + {(y - 13)^2} = 52\)

\({(x - 5)^2} + {(y - 10)^2} = 13\) and \({(x - 7)^2} + {(y - 13)^2} = 52\)

In solving a line with a circle, after substitution the following quadratics are obtained. Decide whether the solution indicates that the line would intersect, touch or miss the circle, and select the correct option.

\[{x^2} - 5x + 6 = 0\]

The lines intersect

The lines touch

The lines miss

\[{x^2} - 6x + 9 = 0\]

\[3{x^2} - 3x + 5 = 0\]

\[3{x^2} + 13x + 4 = 0\]

If the point \(\left( {(p + 1),(p + 2)} \right)\) lies on the circle with equation \({x^2} + {y^2} = 41\), what is \(p\) equal to?

\(3\) or \(- 6\)

\(4\) or \(5\)

\(- 4\) or \(- 5\)

The circle with centre \((2,3)\) and radius \(6\) has what equation?

\[{x^2} + {y^2} + 4x + 6y - 23 = 0\]

\[{x^2} + {y^2} - 4x - 6y - 23 = 0\]

\[{x^2} + {y^2} - 6x - 4y - 23 = 0\]

The circle with centre \(( - 2,5)\) and radius \(\sqrt {29}\) has what equation?

\[{x^2} + {y^2} - 4x + 10y = 0\]

\[{x^2} + {y^2} - 10x + 4y = 0\]

\[{x^2} + {y^2} + 4x - 10y = 0\]

The circle with equation \({x^2} + {y^2} + 14x - 2y + 5 = 0\) has its centre at:

\[(1, - 7)\]

\[(7, - 1)\]

\[( - 7,1)\]

What is the radius of the circle with equation \({x^2} + {y^2} - 8x + 6y + 12 = 0\)?

\[\sqrt {13}\]

\[\sqrt {37}\]

\[\sqrt {88}\]