Equation of normal to the curve at a given point

Jun 14, 2020, 16:45 IST

Equation of normal to the curve at a given point

Definition :-In mathematics the word 'normal' has a very specific meaning. It means 'perpendicular' or at right angles.
If we have a curve such as that shown in the given figure, we can choose a point and draw in the tangent to the curve at that point. The normal is then at right angles to the curve so it is also at right angles (perpendicular) to the tangent. We now find the equation of the normal to a curve.

Note :If two lines, having gradients m1 and m2 respectively, are at right angles to each other then the product of their gradients, m1m2, must equal −1.

Example 1 :-Find the equations of the normal to the curve which are parallel to the line x + 14y + 4 = 0.

Solution:-
equation of the curve is

Slope of the normal at point

On substitution, we get

Normal to the curve is parallel to the line x + 14y + 4 = 0.

So the slope of the line is the slope of the normal.

When x = 2, y = 18 and when x = -2, y =-6
Therefore, there are two normal to the curve

Equation of normal through point (2,18) is given by:

Equation of normal through point (–2, –6) is given by:

Therefore, the equation of normal to the curve are x+14y-254 = 0 And x + 14y + 86 = 0