wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Find the equations of the tangent and normal to the given curves at the indicated points:

(i) y = x4 − 6x3 + 13x2 − 10x + 5 at (0, 5)

(ii) y = x4 − 6x3 + 13x2 − 10x + 5 at (1, 3)

(iii) y = x3 at (1, 1)

(iv) y = x2 at (0, 0)

(v) x = cos t, y = sin t at

Open in App
Solution

(i) The equation of the curve is y = x4 − 6x3 + 13x2 − 10x + 5.

On differentiating with respect to x, we get:

Thus, the slope of the tangent at (0, 5) is −10. The equation of the tangent is given as:

y − 5 = − 10(x − 0)

y − 5 = − 10x

⇒ 10x + y = 5

The slope of the normal at (0, 5) is

Therefore, the equation of the normal at (0, 5) is given as:

(ii) The equation of the curve is y = x4 − 6x3 + 13x2 − 10x + 5.

On differentiating with respect to x, we get:

Thus, the slope of the tangent at (1, 3) is 2. The equation of the tangent is given as:

The slope of the normal at (1, 3) is

Therefore, the equation of the normal at (1, 3) is given as:

(iii) The equation of the curve is y = x3.

On differentiating with respect to x, we get:

Thus, the slope of the tangent at (1, 1) is 3 and the equation of the tangent is given as:

The slope of the normal at (1, 1) is

Therefore, the equation of the normal at (1, 1) is given as:

(iv) The equation of the curve is y = x2.

On differentiating with respect to x, we get:

Thus, the slope of the tangent at (0, 0) is 0 and the equation of the tangent is given as:

y − 0 = 0 (x − 0)

y = 0

The slope of the normal at (0, 0) is , which is not defined.

Therefore, the equation of the normal at (x0, y0) = (0, 0) is given by

(v) The equation of the curve is x = cos t, y = sin t.

∴The slope of the tangent atis −1.

When

Thus, the equation of the tangent to the given curve at is

The slope of the normal atis

Therefore, the equation of the normal to the given curve at is


flag
Suggest Corrections
thumbs-up
1
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Characteristics of Sound Waves
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon