Question

what is normal to a circle ?

Answer 1

it is the line perpendicular to the tangent (where the line meets the curve or circle at one point) and the products of their gradients is equal to minus 1.

E.g. when gradient of tangent is represeted by m1 and gradient of normal is represented by m2...

m1 x m2 = -1.

This is very useful as sometimes you would have the gradient of the tangent and be asked to find the equation of the normal using the co-ordinates (x,y). In order to do this, you must first find the gradient of the normal i.e. m2 = -1 / m1 and then put it into the formula y - y1 = m(x - x1).

Also in circle geometry, the general equation is (x-a)^2 + (y-b)^2 = r^2 where r denotes the radius of a circle and a and b the centre. This is obtained by completing the square and taking all constant terms to the right hand side (r^2 side).

Answer 2

$\huge\bold\red{Definition:-}$

The normal to a curve at a given point is the line perpendicular to the tangent at that point.

In other words, the line perpendicular to the tangent (to a curve), and passing through the point of contact, is known as the normal.

Talking about normals to a circle, there is a peculiar property – the normal will always pass through the center.

Why? Because, in a circle, the radius is always perpendicular to the tangent (at the point of contact). Hence it must coincide with the normal (as per the definition) and therefore the normal must pass through the center.