(x, y) 平面的坐標幾何：練習題

A straight line passes through the point \((1, 2)\) and is parallel to the line with equation \(y = 4x - 5\). Find the equation of this line.

The points \( A(1, 2) \) and \( B(5, 8) \) are the endpoints of a diameter of a circle. Find the equation of this circle.

A circle C has the equation \(x^2 + y^2 - 10x + 4y + k = 0\), where \(k\) is a constant. The line \(l\) with equation \(y = 2x - 1\) is a tangent to the circle C at the point P. Find the coordinates of P and the value of \(k\).

A circle has the equation \( (x + 3)^2 + (y - 5)^2 = 16 \). Find the coordinates of the center and the length of the radius of the circle.

Find the possible values of the constant \(k\) such that the line with equation \(y = 2x + k\) is a tangent to the circle with equation \(x^2 + y^2 = 5\).

A circle is defined by the equation \( (x + 5)^2 + (y - 2)^2 = 49 \).
State the coordinates of the centre of the circle and calculate its radius.

The circle \( C \) has centre \( (3, 4) \). The point \( P(7, 7) \) lies on the circumference of \( C \).
Find the equation of the tangent to the circle at the point \( P \), giving your answer in the form \( ax + by + c = 0 \).

Line \( L_1 \) passes through the points \( (0, 4) \) and \( (2, 0) \).
Find the equation of \( L_1 \) in the form \( ax + by + c = 0 \), where \( a, b, \) and \( c \) are integers.