Equation of a tangent to a circle

A circle has equation (x − 2)² + (y + 1)² = 25.

The point P(5, 3) lies on the circle.

Find an equation of the tangent to the circle at P. Give your answer in the form ax + by = c.

The circle C has equation x² + y² = 25

The point P (3, 4) lies on C.

Find an equation of the tangent to C at the point P.

The circle C has equation x² + y² = 29

The point P (−2, 5) lies on C.

Find an equation of the tangent to C at the point P.

A circle has equation (x + 3)² + (y − 2)² = 13.

The point P(−1, 5) lies on the circle.

Find an equation of the tangent to the circle at P. Give your answer in the form ax + by = c.

A circle has centre O, the origin, and passes through the point T (5, 12).

1. Find an equation of the circle.
2. The point D has coordinates (10, 9). Decide whether D is inside the circle, on the circle, or outside the circle. You must show your working.

The circle C has centre O, the origin, and equation x² + y² = 100

The tangent to C at the point P (6, 8) crosses the x-axis at the point A.

Find the exact coordinates of A.

The circle C has centre O, the origin, and equation x² + y² = 45

A tangent to C touches the circle at the point P.

The tangent has gradient −12 and P is in the first quadrant.

Find the coordinates of P.

A circle has equation x² + y² = 20.

The point A(4, 2) lies on the circle.

1. Find an equation of the tangent to the circle at A.
2. The tangent meets the x-axis at P and the y-axis at Q. Work out the area of triangle OPQ, where O is the origin.

The line with equation 3x + 4y = 30 is a tangent to a circle with centre O, the origin.

Find an equation of the circle.

The centre of a circle is C(1, 2). The point P(4, −2) lies on the circle.

1. Work out the radius of the circle.
2. Find an equation of the tangent to the circle at P. Give your answer in the form ax + by + c = 0, where a, b and c are integers.