Tangent of a curve
1. Tangent: Means touching line
2. Tangent: Let y = f (x) be a curve and P be a point on the curve.
If Q (P) is a point on the curve, then is called a secant line. If the secant line approaches the same limiting position as Q moves along the curve and approaches to P from either side, then limiting position is called a tangent line to the curve at P. The point P is called point of contact of the tangent line to the curve y = f(x).
Note: If P is a point on the curve y = f(x) then there exists all most one tangent at P to the curve y = f(x).
Normal of a curve
Normal: Let y = f(x) be a curve and P be a point on the curve. The line passing through P and perpendicular to the tangent y = f(x) at P is called the normal to the curve y = f(x) at P.
Gradient of a curve: Let y = f(x) be a curve and P be a point on the curve. The slope of the tangent to the curve y = f(x) at P is called gradient to the curve at P.
Angle between two curves
* The acute angle which two curves intersect at a point is defined as the angle between their tangents at the point of intersection.
* If the angle at a point of intersection is a right angle then the two curves are said to cut each other orthogonally at the point of intersection.
* If the angle at a point of intersection is zero then the two curves are said to touch each other at the point of intersection.
* If m1 is the slope of the tangent to the curve f(x) and m2 is the slope of the tangent to the
curve g(x) such that 1 + m1 m2 0 then tan
* If 1 + m1m2 = 0 m1m2 = -1 then the two curves cut each other orthogonal at P.
* If m1 = m2 then the two curves touch each other at P. In this case the two curves have a common tangent and a common normal at P.
* Let us consider a curve whose equation is y = f(x).
* Let P (x1, y1) be a point on the curve.
Let P (x1, y1) be a point on the curve.
* Let the tangent at P be not perpendicular to X -axis.Let the tangent meet the X -axis in T.
* Draw PN perpendicular to the tangent at P to meet in N. Draw
* The length of the segment PT is called the length of the tangent at P.
* The length of the segment PN is called the length of the normal at P.
* TQ, The projection of PT on OX, is called the length of the sub tangent.
* QN, The projection PN on OX, is called the length of the sub normal.