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BodhiAI
Nov. 6, 2019

Normal to Ellipse in Various FOrms,co-normal Points& their Properties:

Equation of Normal in Different Forms

A. Point Form

Let be any point on the ellipse Now, the equation of normal at the point P is the line perpendicular to the tangent at point P on the ellipse. Thus, the equation of the tangent at P is

Slope of tangent at P = mT

Slope of normal at

Now, equation of Normal at point P on the ellipse having slope mT is

 

Problem Solving Trick

If be the general equation of two degree in two variables in x and y representing the equation of ellipse, then the equation of normal at is

B. Slope Form

Equation of Normal at point on the ellipse is 

.......(1)

Let m is the slope of the normal, then

.....(2)

Also, lies on the ellipse, we have 

{using (2)}

and {using (2)}

Hence, the equation of normal in slope form is

Problem Solving Trick

Condition of Normality: 

The line y =  mx + c is normal to the ellipse if

and the coordinates of point of intersection of this normal with that of ellipse is 

C. Parametric Form

Let is any point on the ellipse in parametric form, then the equation of normal at point on the ellipse can be obtained by substituting in the equation of Normal in Point form 

Number of Normals

There can be at most four normals to the ellipse from a single point lying inside the ellipse.

Reasoning: Since, the equation of normal to ellipse at any point on the ellipse in parametric form is

We can have as many normals on the ellipse for every new value of parameter .

Let us assume that the normals intersect at a point P say (h, k), then

which is a fourth degree equation in

Hence, we can have four normals from any point P(h, k)

There can be four normals for any four values of t or tanθ/2, let be those values of which satisfy the above equation. Such that and are the eccentric angles which give the coordinates of co-normal points on the ellipse, through which normals drawn to the ellipse intersect in a point i.e. concurrent.