 # Equation of a Hyperbola referred to Two Perpendicular Lines

Equation of a Hyperbola referred to Two Perpendicular Lines Consider the hyperbola . From figure Let P (x₁, y₁) be any point on the hyperbola. Then, PM = y and PN = x Therefore, . If the perpendicular distances P₁ and P₂ of a moving point P (x, y) from two mutually perpendicular coplanar straight Read more about Equation of a Hyperbola referred to Two Perpendicular Lines[…] # Circle – Director Circle and Its Equation

Circle – Director Circle and Its Equation Director Circle: The locus of the point of intersection of two perpendicular tangents to given circle is known as Director Circle. Equation of Director Circle: Method (1): The equation of any tangent to the circle x² + y² = a² is y = mx + a √ (1 Read more about Circle – Director Circle and Its Equation[…] # Equation of the Parabola when Vertex is (h, k) and Axis is Parallel to X – axis and Y – axis

Equation of the Parabola when Vertex is (h, k) and Axis is Parallel to X – axis and Y – axis Equation of the Parabola when Vertex is (h, k) and Axis is Parallel to X – axis: The parabola y² = 4ax Can be written as (y – 0)² = 4a(x – 0) … Read more about Equation of the Parabola when Vertex is (h, k) and Axis is Parallel to X – axis and Y – axis[…] # Pair of Straight Lines – Homogeneous Second Degree Equation

Pair of Straight Lines – Homogeneous Second Degree Equation An equation (whose RHS is zero) in which the sum of the power of x and y in every term is the same say n, is called a Homogeneous equation of nth degree in x and y. Thus, ax² + 2hxy + by² = 0 is Read more about Pair of Straight Lines – Homogeneous Second Degree Equation[…] # Pair of Straight Lines – Theorem

Pair of Straight Lines – Theorem Combined Equation of Pair of Lines joining Origin and the Point of Intersection of a Curve and A Line Let us find the equation of the straight line joining the origin and the Points of intersection of the curve ax² + 2hxy + by² + 2gx + 2fy + Read more about Pair of Straight Lines – Theorem[…] # Polar Coordinates

Polar Coordinates Polar Coordinates express the location of a point as (r, θ). Where r is the distance from the origin (pole) to the point and θ is the angle from the positive x – axis to the point (in degree or radius). Here, r and θ are called polar coordinates. The distance r is Read more about Polar Coordinates[…] # Shortest Distance between Two Skew Lines

Shortest Distance between Two Skew Lines Definition: l₁ and l₂ are two skew lines. If P is a point on l₁ and Q is a point on l₂ such that PQ perpendicular to l₁ and PQ is called shortest distance and PQ is called shortest distance line between the lines l₁ and l₂. Theorem: The Read more about Shortest Distance between Two Skew Lines[…] # Vector Triple Product

Vector Triple Product The vector triple product of three vectors , and is the vector . Also, . In general, . If , then the vectors and are collinear. is a vector perpendicular to and but is a vector perpendicular to the plane of and . Hence, vector must lie in the plane of and Read more about Vector Triple Product[…] # Applications of Dot (Scalar) Product

Applications of Dot (Scalar) Product Finding Angle between Two Vectors If and are non-zero vectors, then the angle between them is given by . Also . Or . Cosine Rule using Dot Product: Using vector method, prove that in a triangle a² = b² + c² – 2bc cos A (Cosine law). In ΔABC, Let Read more about Applications of Dot (Scalar) Product[…] # Properties of Vector Addition

Properties of Vector Addition 1. Communitive property . 2. Associative property . 3. Additive identity . 4. Additive inverse . 5. and . Examples 1: If vector bisects the angle between and , then prove that . Solution: We know that vector is along the diagonal of the parallelogram whose adjacent sides are vectors and Read more about Properties of Vector Addition[…]