# Â  Â  Â  Â  Â EXERCISE 6.3

Question 1: Find the slope of the tangent to the curve at .

Solution: The slope of tangent to the curve

If , then slope

Question 2: Find the slope of the tangent to the curve at .

Solution: The slope of tangent to the curve

If , then slope

Question 3: Find the slope of the tangent to curve at the point whose -coordinate is 2 .

Solution: The slope of tangent to the curve

If -coordinate is 2 , then slope

Question 4: Find the slope of the tangent to the curve at the point whose -coordinate is 3 .

Solution: The slope of tangent to the curve

If -coordinate is 3 , then slope

Question 5: Find the slope of the normal to the curve at .

Solution: Here,

and

The slope of normal to the curve

If , then slope of normal

Question 6: Find the slope of the normal to the curve at .

Solution: Here,

and

The slope of normal

If , then the slope of normal

Question 7: Find points at which the tangent to the curve is parallel to the -axis.

Solution: The slope of tangent to the curve

If the tangent is parallel to – axis, then slope

If , then

,

so, the points

If , then

,

so, the point

Hence, on the points and , the tangents are parallel to – axis.

Question 8: Find a point on the curve at which the tangent is parallel to the chord joining the points and .

Solution: The slope of tangent to the curve

The slope of line joining the points and

Given: The slope of tangent to the curve slope of line joining the points and .

,

If , then ,

therefore, the required point

Question 9: Find the point on the curve at which the tangent is .

Solution: Given

Slope of tangent to the curve

Slope of the line

.

Given that: The slope of tangent to the curve = the slope of the line

If , then

,

therefore, the point

If , then

,

therefore, the point

Out of the points and , only satisfy the equation of line .

Hence, is the point on at which tangent is .

Question 10: Find the equation of all lines having slope that are tangents to the curve .

Solution: curve

The slope of tangent to the curve

Given that: The slope of tangent to curve is .

If , then

,

therefore, the point .

Hence, the tangent:

If , then

,

therefore, the point .

Hence, the tangent:

Question 11: Find the equation of all lines having slope 2 which are tangents to the curve .

Solution: curve

The slope of tangent to the curve

Given that: The slope of tangent to the curve is 2 .

The square of a real number can’t be negative. Hence, such tangents doesn’t exists.

Question 12: Find the equations of all lines having slope 0 which are tangent to the curve .

Solution: curve

The slope of tangent to the curve

Given that: The slope of tangent to the curve is 0 .

If , then

,

therefore, the point .

Hence, the tangent:

Question 13: Find points on the curve at which the tangents are

(i) parallel to -axis

(ii) parallel to -axis.

Solution:

Differentiating with respect to ,

The slope of tangent

(i) If the tangent is parallel to – axis, then slope of tangent

Putting the value of in ,

Therefore, the required points are and .

(ii) If the tangent is parallel to -axis, the slope of normal

Putting the value of in , we have

Therefore, the required points are and .

Question 14: Find the equations of the tangent and normal to the given curves at the indicated points:

(i) at

(ii) at

(iii) at

(iv) at

(v) at

Solution: (i) curve

The slope of tangent to the curve

The slope of tangent at is

Therefore, the equation of tangent passing through is

The slope of normal at is

Therefore, the equation of normal through is

(ii) curve

The slope of tangent to the curve

The slope of tangent at is

Therefore, the tangent at is

The slope of normal at is

The equation of normal through is

(iii) curve

The slope of tangent to the curve

The slope of tangent at is

Therefore, the equation of tangent through is

Slope of normal at is

Therefore, the equation of normal at is

(iv) curve

The slope of tangent to the curve

The slope of tangent at is

Therefore, the equation of tangent through is

The slope of normal at is

Therefore, the equation of normal at is

(v) curve

The slope of tangent

The slope of tangent at is

Therefore, the equation of tangent at is

The slope of normal at is

Therefore, the equation of normal at is

Question 15: Find the equation of the tangent line to the curve which is

(a) parallel to the line

(b) perpendicular to the line .

Solution: curve

The slope of tangent

(i) The slope of the line

If the tangent to is parallel to , then the slopes are equal

Putting in curve ,

we have, .

So, the point

Therefore, the equation of tangent at is given by

(ii) The line

Slope the line

If the tangent of is perpendicular to , then

we have

Putting in ,

we have

,

Therefore, the point

the slope of tangent

The equation of tangent at is given by

Question 16: Show that the tangents to the curve at the points where and are parallel.

Solution: curve

The slope of tangent

The slope of tangent at

The slope of tangent at

The slope of tangent at and is same.

Hence, the tangents are parallel to each other.

Question 17: Find the points on the curve at which the slope of the tangent is equal to the -coordinate of the point.

Solution : The curve is

The slope of tangent

Let be any point on ,

–(i)

where the slope of tangent is equal to – coordinate of the point.

–(ii)

Solving the equation (i) and (ii), we have,

From the equation (1), we have

If , then ,

therefore, the required point

If , then ,

therefore, the required point

Question 18: For the curve , find all the points at which the tangent passes through the origin.

Solution: The curve

The slope of tangent

Let, the point lies on the curve ,

—(i)

The slope of tangent at

—(ii)

Slope of the line through and is given by –(iii)

From the equation (ii) and (iii), we have

—(iv)

Solving the equation (1) and (iv), we have

From the equation (iv), we have

If , then

,

therefore, the required point

If , then

,

therefore, the required point

If , then

,

therefore, the required point

Question 19: Find the points on the curve at which the tangents are parallel to the -axis.

Solution:
Differentiating with respect to x:

The slope of tangent

The tangent is parallel to -axis.

Therefore,

Putting in the curve

, we have

Therefore, the points are and .

The tangent at point is given by

The tangent at point is given by

Question 20: Find the equation of the normal at the point for the curve .

Solution :

Differentiating with respect to , we have

The slope of normal to the curve

The slope of normal at is given by

The equation of normal at

Question 21: Find the equation of the normals to the curve which are parallel to the line .

Solution:

Differentiating with respect to , we have

The slope of normal to is

The slope of line is

If the normal to is parallel to line , the both have the same slope.

Therefore,

Putting in , we have,

,

therefore the point

The equation of normal at is given by

Putting in , we have,

,

therefore, point

Therefore, the equation of normal at is given by

Question 22: Find the equations of the tangent and normal to the parabola at the point .

Solution :

Differentiating with respect to , we have

The slope of normal to is given by

Therefore, the slope of normal at

The equation of normal at

Question 23: Prove that the curves and cut at right angles if .

Solution: –(i)

–(ii)

Putting the value of

Solving (i) and (ii)

Putting the value of in equation (1), we have,

Hence, the two curves intersects at the point .

Differentiating with respect to , we have

The curve

The slope of tangent

Slope of tangent at point is given by

Differentiating with respect to , we have

The slope of tangent to the curve is

The slope of tangent at is given by

The curves and intersect each other at right angle.

Question 24: Find the equations of the tangent and normal to the hyperbola at the point .

Solution: Since

Differentiating with respect to , we have

The slope of tangent to hyperbola is

Therefore, the slope of tangent at is given by

The equation of tangent at :

The slope of normal to hyperbola is given by

The slope of normal at is given by

The equation of normal at :

Question 25: Find the equation of the tangent to the curve which is parallel to the line .

Solution: The curve

The slope of tangent is

The slope of line

If the tangent to is parallel to , then their slopes will be equal.

Therefore,

Putting in the equation , we have,

Therefore, the point

Therefore, the equation of tangent at :

Choose the correct answer in Exercises 26 and 27.

Question 26: The slope of the normal to the curve at is

(A) 3

(B)

(C)

(D)

Solution: Differentiating with respect to , we have,

Slope of normal to is given by

The equation of normal at

Hence, the option (D) is correct.

Question 27: The line is a tangent to the curve at the point

(A)

(B)

(C)

(D)

Solution: Differentiating with respect to , we have

The slope of tangent to the curve is

Slope of the line is given by

If the tangent of is .

Therefore,

Putting in , we have,

.

Therefore, the point

Hence, the option (A) is correct.