By Team Aakash Byju's | 27th January 2023

How to Find Points on the Curve Whose Tangents are Parallel to Axes?

Find points on the curve

Parallel to the x-axis.

The points at which the tangents are parallel to the x-axis are (0,4) and (0,−4).

The points at which the tangents are parallel to the  y-axis are (3,0) and (−3,0).

A

B

C

x

at which the tangents are:

9

+

y

16

2

2

= 1

Given,

differentiating this with respect to x on both sides, we get

x

9

+

y

16

2

2

= 1

2x

9

+

2y

16

= 0

9y

-16x

=>

dx

dy

dx

dy

x

y

(3,0)

(-3,0)

(0,4)

(0,-4)

Now, for a tangent to be parallel to the x-axis, its slope has to be zero because it will be a horizontal line.

x

y

9y

-16x

=>

dx

dy

=0

(3,0)

(-3,0)

(0,4)

(0,-4)

But this is possible only if x=0

Then

x

9

y

16

2

2

= 1

+

for x=0, => y   =16 => y=   4

2

+

-

Therefore the points are (0, 4)  and (0, −4)

x

y

(3,0)

(-3,0)

(0,4)

(0,-4)

Now, for a tangent to be parallel to the y-axis, the slope of its normal has to be zero.

1

dy

-

= 0

dx

-1

-16

9y

(

)

=

16x

9y

= 0

=> y=0

x

y

(3,0)

(-3,0)

(0,4)

(0,-4)

Then ,

x

9

+

y

16

2

2

= 1

Therefore the points are (3, 0)  and (−3, 0)

for y=0, => x=    3

+

-

x

y

(3,0)

(-3,0)

(0,4)

(0,-4)