Question

Let L be a normal to the parabola y² = 4x. If L passes through the point (9,6),then write the equation of L.

IiT-JEE 2011​

Answer 1

Given

The parabola y² = 4x

To Find

The equation of normal passing through (9,6).

Calculations

We know, normal to any parabola is given by y = mx - 2am - am³.

Here,for the given parabola y² = 4x.

4a = 4

=> a = 1.

Putting the value of a in the equation of normal.

y = mx - 2(1)m -(1)m³

=> y = mx - 2m - m³ .....(i)

Now,this normal passes through (9,6). Hence, putting x = 9 and y = 6,we get :-

=> 6 = 9m - 2m - m³

=> m³ - 7m - 6 = 0

=> m³ + m² - m² - m - 6m - 6 = 0

=> m²(m + 1) - m(m + 1) - 6(m + 1) = 0

=> (m + 1) ( m² - m - 6) = 0

=> (m + 1) (m+3) ( m - 2) = 0.

We get, m = -3, m = -2 and m = 2. Put this value in eqn(i) to get the equation of L.

putting m = -3

=> y = -3x - 2(-3) - (-3)³

=> y = -3x + 33 ......first equation of L

putting x = -1

=> y = - x -2(-1) -(-1)³

=> y = - x + 3 ....... second equation of L

putting x = 2

=> y = 2x - 2(2) - (2)³

=> y = 2x - 12....third equation of L.

Answer 2

Step-by-step explanation:

Given

The parabola y² = 4x

To Find

The equation of normal passing through (9,6).

Calculations

We know, normal to any parabola is given by y = mx - 2am - am³.

Here,for the given parabola y² = 4x.

4a = 4

=> a = 1.

Putting the value of a in the equation of normal.

y = mx - 2(1)m -(1)m³

=> y = mx - 2m - m³ ..(i)

Now,this normal passes through (9,6). Hence, putting x = 9 and y = 6,we get :-

=> 6 = 9m - 2m - m³

=> m³ - 7m - 6 = 0

=> m³ + m² - m² - m - 6m - 6 = 0

=> m²(m + 1) - m(m + 1) - 6(m + 1) = 0

=> (m + 1) ( m² - m - 6) = 0

=> (m + 1) (m+3) ( m - 2) = 0.

We get, m = -3, m = -2 and m = 2. Put this value in eqn(i) to get the equation of L.

putting m = -3

=> y = -3x - 2(-3) - (-3)³

=> y = -3x + 33 ...first equation of L

putting x = -1

=> y = - x -2(-1) -(-1)³

=> y = - x + 3 .. second equation of L

putting x = 2

=> y = 2x - 2(2) - (2)³

=> y = 2x - 12...third equation of L.

hope this helps you!!