Let X’OX, Y’OY and Z’OZ be three mutually perpendicular lines intersecting at O such that two of them viz. Y’OY and Z’OZ lie in the plane of the paper and the third X’OX is perpendicular to the plane of the paper and is projecting out from the plane of the paper (see Fig shown below.). Read more about Co – ordinates of point in space[…]

Let (x1, y1) be the point of intersection of the lines represented by the given equation. Shifting the origin at (x1, y1), we have x = X + x1 and y = Y + y1 … (i) Putting the values of x and y in the given equation, we obtain a (X + x1)2 + Read more about Point of Intersection of Pair of Straight Lines[…]

Condition for the lines to be coincident: The lines are coincident if the angle between them is zero. Thus, lines are coincident. ⇒ θ = 0 ⇒ tanθ = 0 ⇒ . ⇒ h2 – ab = 0 ⇒ h2 = ab Hence, the lines represented by ax2 + 2hxy + by2 = 0 are Read more about Lines to be Coincident, Perpendicular & Equation of Second Degree & Bisectors[…]

Let y = m1 x and y = m2 x be the lines represented by ax2 + 2hxy + by2 = 0. Then, and … (i)Let θ be the angle between the lines y = m1 x and y = m2 x. Then, ⇒ . ⇒ . ⇒  [Using (i)] ⇒ , ⇒ . Thus, Read more about ANGLE BETWEEN THE PAIR OF LINES[…]

Normal or perpendicular form of a line: The equation of the straight line upon which the length of the perpendicular from the origin is P and this perpendicular makes an angle α with x – axis is x cosα + y sinα = P. Example: A line forms a triangle of area 54√3 sq units Read more about Various forms of equation of straight lines – I[…]

A straight line is a curve such that every point on the line segment joining any two points on it lies on it. Slope (Gradient) of a line: The trignometrical tangent of the angle that a line makes with the positive direction of the x – axis in anticlockwise sense is called the slope or Read more about Various forms of equation of straight lines[…]

Lines parallel and perpendicular to a given line: 1. The equation of line parallel to a given line ax + by + c = 0 is ax + by + λ = 0 where λ is constant. 2. The equation of line perpendicular to a given line ax + by + c = 0 is Read more about Lines through Points of Intersection of Two Lines[…]

Equation of the Bisectors: The bisectors of the angle between two straight lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are the locus of a point which is equidistant from the two lines. So let P (h, k) be a point equidistant from the two lines. Then, Read more about Equations of the Bisectors of the Angle between Lines[…]

The length of the perpendicular from a point (x₁, y₁) to a line ax + by + c = 0 is . The length of the perpendicular from the origin to the line ax + by + c = 0 is . Example: If P is the length of the perpendicular from the origin to Read more about Distance of a Point from a Line[…]

Centroid: The centroid of a triangle is the point of intersection of medians. It divides medians in 2: 1 ratio. If A(x1, y1), B(x2, y2), C(x3, y3) are vertices of triangle ABC, then coordinates of centroid is .In center: Point of intersection of angular bisectors Coordinates of . Where a, b, c are sides of triangle Read more about Centroid, Circumcentre, Orthocentre, Incentre of Triangle[…]