Coordinate Geometry

Angle between Two Lines

Let us consider, the two lines \(y_1 = m_1 x + c_1\) and \(y_2 = m_2 x + c_2\) with \(\theta\) being the angle between them.

Key Note: 

- The \(\pm\) accounts for the two possible orientations of the angle between the lines (acute or obtuse). \(-\) for Acute Angle and \(+\) for Obtuse Angle

Special Conditions

1. Lines being Parallel

- \(m_1 = m_2\) or \(\theta = 0^0\) 

2. Lines being Perpendicular

- \(m_1 m_2 = -1\) or \(\theta = 180^0\) 

Pair of Straight Lines

The standard/combined form for the pair of straight lines through the origin is given as \(ax^2 + 2hxy + by^2 = 0\). Let \(\theta\)  be the angle between the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\), then,

Key Note: 

- The \(\pm\) accounts for the two possible orientations of the angle between the lines (acute or obtuse). \(-\) for Acute Angle and \(+\) for Obtuse Angle

Special Conditions

1. Lines being Parallel (\(\theta = 0^0\)): \(h^2 = ab\)

2. Lines being Perpendicular (\(\theta = 90^0\)): \(a + b = 0\) 

Circle

A circle is a two-dimensional shape where every point on its edge is equally distant from a central point, known as the center. This distance from the center to any point on the circumference is called the radius of the circle.

Some Important Formulas

1. Equation of Circle with center \((h, k)\) and radius \(r\)

2. If the endpoints of the diameter are given \((x_1, y_1)\) and \((x_2, y_2)\)

3. If the equation of a circle is \(x^2 + y^2 + 2gx + 2fy + c = 0\)