Normal to a Parabola

What do we mean by a normal to a parabola?

The concept of normal is closely related to that of tangents. The product of the slopes of tangent and normal is equal to -1. This concept is extremely important and it often fetches questiosn in competitive exams like the JEE. The normals to a parabola can be written in various forms, i.e. there can be various equations of normals to a parabola. We discuss the various forms one by one:  

Normal at the point (x₁, y₁) 

The equation of the tangent at the point (x₁, y₁) is yy₁ = 2a(x + x₁).If the slope of the tangent = 2a/y₁ then the slope of the normal is –y₁/2a. Also it passes through (x₁, y₁). 

Hence, its equation is y – y₁ = -y₁/2a (x – x₁). … (i)

Normal in terms of slope (m) 

In equation (i), put -y₁/2a = m so that y₁ = –2a and x₁ = (y₁²)/4a = am², then the equation becomes y = mx – 2am – am³ where m is a parameter. Equation (ii) is the normal at the point (am², –2am) of the parabola. 

Remark: 

• If this normal passes through a point (h, k), then k = mh – 2am – am³.

• For a given parabola and a given point (h, k), this cubic in m has three roots say m₁, m₂, m₃ i.e. from (h, k) three normals can be drawn to the parabola whose slopes are m₁, m₂, m₃. For this cubic, we have m₁ + m₂ + m₃ = 0, m₁ m₂ + m₂ m₃ + m₃ m₁ = (2a – h)/a and m₁ m₂ m₃ = –k/a.

• If we have an extra condition about the normals drawn from a point (h, k) to a given parabola y² = 4ax then by eliminating m₁, m₂, m₃ from these four relations between m₁, m₂, m₃, we can get the locus of (h, k). 

• Since the sum of the roots is equal to zero, the sum of the ordinates of the feet of the normals from a given point is zero. These points are called Co-Normal Points.

Normal at the point t

The normal, being perpendicular to the tangent at (at, 2at) is given by y = –tx + 2at + at³. 

Remark:

If normal at the point at₁ meets the parabola again at the point at₂, then at₂ = – at₁ – 2/at₁ .

Point of intersection of the normals to the parabola y² = 4ax t₁ and t₂ are (2a + a(t₁² + t₂² + t₁t₂), -a t₁t₂(t₁+t₂)).
 

Properties of a normal to a Parabola y² = 4ax

• From a given point, a total of three normals can be drawn to the parabola y² = 4ax.

• The algabraic sum of the slope of these normals is zero.

• The algebraic sum of the ordinates of their feet is zero.

• The centroid of the triangle formed by joining their feet lies on the axis of the parabola.

• The tangent at one extremity of a focal chord of a parabola is parallel to the normal at the other extremity.

• The normals lying on the extremities of the latus rectum of a parabola intersect at right angles on the xais of the parabola.

• Atleast one of the normal is real since imaginary normals always occur in pairs.

• The length of the normal is gievn by the formula a(t₁ - t₂)(t₁ + t₂)² + 4 = 4a(t₁² + 1)^3/2 /t₁²

• When the feet of the normals of the parabola are joined, they form a triangle whose centroid is given by the formula ((am₁² + am₂² + am₃²)/3, (2am₁ + 2am₂ + 2am₃)/ 3) which is equal to (am₁² + am₂² + am₃²) /3, 0)

We now discuss some of the illustrations based on the above concepts:

Illustration: 

Find the locus of the point of intersection of two normals to a parabola which are at right angles to one another.

Solution:

The equation of the normal to the parabola y² = 4ax is y = mx – 2am – am³.

It passes through the point (h, k) if 

k = mh – 2am – am³ => am³ + m(2a – h) + k = 0. … (1)

Let the roots of the above equation be m₁, m₂and m_3. Let the perpendicular normals correspond to the values of m₁ and m₂ so that m₁m₂ = –1.

From equation (1), m₁ m₂ m₃ = -k/a. Since m₁ m₂ = –1, m₃ = k/a.

Since m3 is a root of (1), we have a (k/a)³+k/a (2a – h) + k = 0. ⇒ k² + a(2a – h) + a² = 0

⇒ k² = a(h – 3a).

Hence the locus of (h, k) is y² = a(x – 3a).

Illustration:
Prove that the normal chord to a parabola at the point whose ordinate is equal to the abscissa subtends a right angle at the focus.

Solution:

If the normal to the parabola y² = 4ax at P(at₁t₂, 2at₁) meets it again at the point t₂, then we have t₂ = – t₁ – 2/t₁ 

If the abscissa and the ordinates of P be equal, then at₁² = 2at₁

⇒ t₁ = 2 (rejecting t₁ = 0) ? t₂ = – 2 – 1 = – 3

The co-ordinates of P and Q are therefore (4a, 4a) and (9a, – 6a) respectively.

The focus is the point S (a, 0).

Slope of PS = 3/4 and slope of QS = – 3/4.

⇒ ∠PSQ = right angle. Hence the result.

Illustration:

Find the locus of the middle points of the normal chords of the parabola y² = 4ax. 

Solution:

Equation of the normal chord at any point (at₂, 2at) of the parabola is

y + tx = 2at + at³. … (1)

Equation of the chord with mid point (x₁, y₁) is T = S1

or yy₁ – 2a(x + x₁) = y₁² – 4ax₁ or yy₁ – 2ax = y₁² – 2ax₁. … (2)

Since equations (1) and (2) are identical, 1/y₁ = t /(-2a) = (2at + at³)/t = 2a + ((-2a)/y₁)²

or -(y₁²) /2a + x₁ = 2a + 4a³ /(y₁²) or x₁ – 2a = (y₁²) /2a + 4a³ /(y₁²)

Hence the locus of the middle point (x₁, y₁) is x – 2a = y²/2a + 4a³ /y² .

Illustration: 

P and Q are the points t₁ and t₂ on the parabola y² = 4ax. If the normals to the parabola at P and Q meet at R, (a point on the parabola), show that t₁t₂ = 2. 

Solution:

Let the normals at P and Q meet at R(at₂, 2at).

Then t = – t₁ – 2/t₁ and t = – t₂ – 2/t₂ .

Therefore t₁ + 2/t₁ = t₂ + 2/t₂ ⇒ (t₁ – t₂) = 2(t₁-t₂ )/(t₁ t₂ ) ⇒ t₁t₂ = 2.

Illustration:

Find the equations of the normals to the parabola y² = 4ax at the extremities of its latus rectum. If the normals meet the parabola, again at P and Q, prove that PQ = 12a. 

Solution: 

The ends of the latus rectum are (a, 2a) and (a, –2a). The equations of the normals to the parabola at these points are (put t = 1 and –1)

y + x = 3a and y – x = 3a.

These lines meet the parabola again at P(9a, 6a) and Q(9a, 6a) respectively.

⇒ PQ = 6a + 6a = 12a.
 

Related Resources

• Click here to refer the most Useful Books of Mathematics.

• For getting an idea of the type of questions asked, refer the previous year papers.

To read more, Buy study materials of Parabola comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.