## Exam-Style Questions.## Problems adapted from questions set for previous Mathematics exams. |

## 1. | GCSE Higher |

What is the equation of a circle, centered at the origin, that has a tangent passing through the points (-30, 0) and (0, -15)?

Show all of your working.

## 2. | GCSE Higher |

The diagram shows a circle with equation \(x^2+y^2=13\).

A tangent to the circle touches the circle at the point J. The x-coordinate of J is 2. The tangent intersects the x-axis at K. Find the coordinates of the point K.

## 3. | GCSE Higher |

(a) Find the coordinates of the point at which the curve \(y = k^x\) intersects the y-axis.

The equation of circle \(A\) is \(x^2+y^2=25\). This circle is translated by the vector \( \begin{pmatrix} 0 \\ 4 \\ \end{pmatrix} \) to give circle \(B\).

(b) Draw a sketch of circle \(B\) clearly labelling the points of intersection with the y-axis.

## 4. | GCSE Higher |

A circle with equation \(x^2 + y^2 = 6 \) meets a one of its tangents at point \(S\).

The coordinates of \(S\) are:

$$\left( \frac{\sqrt 5}{3} , \frac{7}{3} \right)$$Find the equation of the tangent to the circle at the point \(S\).

## 5. | GCSE Higher |

(a) The circumference of a circle is \(16 \pi \) cm and its centre is at the origin. Find the equation of the circle.

(b) The line \(12x+ ay = b\) is a tangent at the point (6, 5) to a different circle with centre at the origin. Find the values of a and b.

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