## EXERCISE 6.3

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#### Question 1:

Find the slope of the tangent to the curve *y* = 3*x*^{4} − 4*x* at *x* = 4.

#### Answer:

The given curve is *y* = 3*x*^{4} − 4*x*.

Then, the slope of the tangent to the given curve at *x* = 4 is given by,

#### Question 2:

Find the slope of the tangent to the curve, *x* ≠ 2 at *x* = 10.

#### Answer:

The given curve is.

Thus, the slope of the tangent at *x* = 10 is given by,

Hence, the slope of the tangent at *x* = 10 is

#### Question 3:

Find the slope of the tangent to curve *y* = *x*^{3} − *x *+ 1 at the point whose *x*-coordinate is 2.

#### Answer:

The given curve is.

The slope of the tangent to a curve at (*x*_{0}, *y*_{0}) is.

It is given that *x*_{0} = 2.

Hence, the slope of the tangent at the point where the *x*-coordinate is 2 is given by,

#### Question 4:

Find the slope of the tangent to the curve *y* = *x*^{3} − 3*x* + 2 at the point whose *x*-coordinate is 3.

#### Answer:

The given curve is.

The slope of the tangent to a curve at (*x*_{0}, *y*_{0}) is.

Hence, the slope of the tangent at the point where the *x*-coordinate is 3 is given by,

#### Question 5:

Find the slope of the normal to the curve *x* = *a*cos^{3}*θ*, *y* = *a*sin^{3}*θ* at.

#### Answer:

It is given that* x* = *a*cos^{3}*θ* and *y* = *a*sin^{3}*θ***.**

Therefore, the slope of the tangent at is given by,

Hence, the slope of the normal at

#### Question 6:

Find the slope of the normal to the curve *x* = 1 − *a *sin *θ*, *y* = *b *cos^{2}*θ* at .

#### Answer:

It is given that *x* = 1 − *a *sin *θ* and *y* = *b *cos^{2}*θ*.

Therefore, the slope of the tangent at is given by,

Hence, the slope of the normal at

#### Question 7:

Find points at which the tangent to the curve *y* = *x*^{3} − 3*x*^{2} − 9*x* + 7 is parallel to the *x*-axis.

#### Answer:

The equation of the given curve is

Now, the tangent is parallel to the *x*-axis if the slope of the tangent is zero.

When *x* = 3, *y* = (3)^{3} − 3 (3)^{2} − 9 (3) + 7 = 27 − 27 − 27 + 7 = −20.

When *x* = −1, *y* = (−1)^{3} − 3 (−1)^{2} − 9 (−1) + 7 = −1 − 3 + 9 + 7 = 12.

Hence, the points at which the tangent is parallel to the *x*-axis are (3, −20) and

(−1, 12).

#### Question 8:

Find a point on the curve *y* = (*x* − 2)^{2} at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

#### Answer:

If a tangent is parallel to the chord joining the points (2, 0) and (4, 4), then the slope of the tangent = the slope of the chord.

The slope of the chord is

Now, the slope of the tangent to the given curve at a point (*x*, *y*) is given by,

Since the slope of the tangent = slope of the chord, we have:

Hence, the required point is (3, 1).

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#### Question 9:

Find the point on the curve *y* = *x*^{3} − 11*x* + 5 at which the tangent is *y* = *x* − 11.

#### Answer:

The equation of the given curve is *y* = *x*^{3} − 11*x* + 5.

The equation of the tangent to the given curve is given as *y* = *x* − 11 (which is of the form *y* = *mx* + *c)*.

∴Slope of the tangent = 1

Now, the slope of the tangent to the given curve at the point (*x*, *y*) is given by,

Then, we have:

When *x* = 2, *y* = (2)^{3} − 11 (2) + 5 = 8 − 22 + 5 = −9.

When *x* = −2, *y* = (−2)^{3} − 11 (−2) + 5 = −8 + 22 + 5 = 19.

Hence, the required points are (2, −9) and (−2, 19). But, both these points should satisfy the equation of the tangent as there would be point of contact between tangent and the curve. ∴ (2, −9) is the required point as (−2, 19) is not satisfying the given equation of tangent.

#### Question 10:

Find the equation of all lines having slope −1 that are tangents to the curve .

#### Answer:

The equation of the given curve is.

The slope of the tangents to the given curve at any point (*x*, *y*) is given by,

If the slope of the tangent is −1, then we have:

When *x* = 0, *y* = −1 and when *x* = 2, *y* = 1.

Thus, there are two tangents to the given curve having slope −1. These are passing through the points (0, −1) and (2, 1).

∴The equation of the tangent through (0, −1) is given by,

∴The equation of the tangent through (2, 1) is given by,

*y* − 1 = −1 (*x* − 2)

⇒ *y* − 1 = − *x* + 2

⇒ *y* + *x* − 3 = 0

Hence, the equations of the required lines are *y* + *x* + 1 = 0 and *y* + *x* − 3 = 0.

#### Question 11:

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

#### Answer:

The equation of the given curve is.

The slope of the tangent to the given curve at any point (*x*, *y*) is given by,

If the slope of the tangent is 2, then we have:

Hence, there is no tangent to the given curve having slope 2.

#### Question 12:

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

#### Answer:

The equation of the given curve is.

The slope of the tangent to the given curve at any point (*x*, *y*) is given by,

If the slope of the tangent is 0, then we have:

When *x* = 1,

∴The equation of the tangent throughis given by,

Hence, the equation of the required line is

#### Question 13:

Find points on the curve at which the tangents are

(i) parallel to *x*-axis (ii) parallel to *y*-axis

#### Answer:

The equation of the given curve is.

On differentiating both sides with respect to *x*, we have:

(i) The tangent is parallel to the *x*-axis if the slope of the tangent is i.e., 0 which is possible if *x* = 0.

Then, for *x* = 0

Hence, the points at which the tangents are parallel to the *x*-axis are

(0, 4) and (0, − 4).

(ii) The tangent is parallel to the *y*-axis if the slope of the normal is 0, which gives⇒ *y* = 0.

Then, for *y* = 0.

Hence, the points at which the tangents are parallel to the *y*-axis are

(3, 0) and (− 3, 0).

#### Question 14:

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

(i) *y* = *x*^{4} − 6*x*^{3} + 13*x*^{2} − 10*x* + 5 at (0, 5)

(ii) *y* = *x*^{4} − 6*x*^{3} + 13*x*^{2} − 10*x* + 5 at (1, 3)

(iii) *y* = *x*^{3} at (1, 1)

(iv) *y* = *x*^{2} at (0, 0)

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

#### Answer:

(i) The equation of the curve is* y* = *x*^{4} − 6*x*^{3} + 13*x*^{2} − 10*x* + 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 = − 10*x*

⇒ 10*x* + *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* = *x*^{4} − 6*x*^{3} + 13*x*^{2} − 10*x* + 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* = *x*^{3}.

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* = *x*^{2}.

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 (*x*_{0, }*y*_{0}) = (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

#### Question 15:

Find the equation of the tangent line to the curve *y* = *x*^{2} − 2*x* + 7 which is

(a) parallel to the line 2*x* − *y* + 9 = 0

(b) perpendicular to the line 5*y* − 15*x* = 13.

#### Answer:

The equation of the given curve is.

On differentiating with respect to *x*, we get:

(a) The equation of the line is 2*x* − *y* + 9 = 0.

2*x* − *y* + 9 = 0 ⇒ *y* = 2*x *+ 9

This is of the form *y* = *mx *+ *c*.

∴Slope of the line = 2

If a tangent is parallel to the line 2*x* − *y* + 9 = 0, then the slope of the tangent is equal to the slope of the line.

Therefore, we have:

2 = 2*x* − 2

Now, *x* = 2

*y* = 4 − 4 + 7 = 7

Thus, the equation of the tangent passing through (2, 7) is given by,

Hence, the equation of the tangent line to the given curve (which is parallel to line 2*x* − *y* + 9 = 0) is.

(b) The equation of the line is 5*y* − 15*x* = 13.

5*y* − 15*x* = 13 ⇒

This is of the form *y* = *mx *+ *c*.

∴Slope of the line = 3

If a tangent is perpendicular to the line 5*y* − 15*x* = 13, then the slope of the tangent is

Thus, the equation of the tangent passing throughis given by,

Hence, the equation of the tangent line to the given curve (which is perpendicular to line 5*y* − 15*x* = 13) is.

#### Question 16:

Show that the tangents to the curve *y* = 7*x*^{3} + 11 at the points where *x* = 2 and *x* = −2 are parallel.

#### Answer:

The equation of the given curve is* y* = 7*x*^{3} + 11.

The slope of the tangent to a curve at (*x*_{0}, *y*_{0}) is. Therefore, the slope of the tangent at the point where *x* = 2 is given by,

It is observed that the slopes of the tangents at the points where *x* = 2 and *x* = −2 are equal.

Hence, the two tangents are parallel.

#### Question 17:

Find the points on the curve *y* = *x*^{3} at which the slope of the tangent is equal to the *y*-coordinate of the point.

#### Answer:

The equation of the given curve is *y* = *x*^{3}.

The slope of the tangent at the point (*x*, *y*) is given by,

When the slope of the tangent is equal to the *y*-coordinate of the point, then *y* = 3*x*^{2}.

Also, we have *y* = *x*^{3}.

∴3*x*^{2} = *x*^{3}

⇒ *x*^{2} (*x* − 3) = 0

⇒ *x* = 0, *x* = 3

When *x* = 0, then *y* = 0 and when *x* = 3, then *y* = 3(3)^{2} = 27.

Hence, the required points are (0, 0) and (3, 27).

#### Question 18:

For the curve *y* = 4*x*^{3} − 2*x*^{5}, find all the points at which the tangents passes through the origin.

#### Answer:

The equation of the given curve is *y* = 4*x*^{3} − 2*x*^{5}.

Therefore, the slope of the tangent at a point (*x*, *y*) is 12*x*^{2} − 10*x*^{4}.

The equation of the tangent at (*x*, *y*) is given by,

When the tangent passes through the origin (0, 0), then *X* = *Y* = 0.

Therefore, equation (1) reduces to:

Also, we have

When *x* = 0, *y* =

When *x* = 1, *y* = 4 (1)^{3} − 2 (1)^{5} = 2.

When *x* = −1, *y* = 4 (−1)^{3} − 2 (−1)^{5} = −2.

Hence, the required points are (0, 0), (1, 2), and (−1, −2).

#### Question 19:

Find the points on the curve *x*^{2} + *y*^{2} − 2*x* − 3 = 0 at which the tangents are parallel to the *x*-axis.

#### Answer:

The equation of the given curve is *x*^{2} + *y*^{2} − 2*x* − 3 = 0.

On differentiating with respect to *x*, we have:

Now, the tangents are parallel to the *x*-axis if the slope of the tangent is 0.

But, *x*^{2} + *y*^{2} − 2*x* − 3 = 0 for *x* = 1.

*y*^{2} = 4 ⇒

Hence, the points at which the tangents are parallel to the *x*-axis are (1, 2) and (1, −2).

#### Question 20:

Find the equation of the normal at the point (*am*^{2}, *am*^{3}) for the curve *ay*^{2} = *x*^{3}.

#### Answer:

The equation of the given curve is *ay*^{2} = *x*^{3}.

On differentiating with respect to *x*, we have:

The slope of a tangent to the curve at (*x*_{0}, *y*_{0}) is.

The slope of the tangent to the given curve at (*am*^{2}, *am*^{3}) is

∴ Slope of normal at (*am*^{2}, *am*^{3}) =

Hence, the equation of the normal at (*am*^{2}, *am*^{3}) is given by,

*y* − *am*^{3} =

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#### Question 21:

Find the equation of the normals to the curve *y* = *x*^{3} + 2*x *+ 6 which are parallel to the line *x* + 14*y* + 4 = 0.

#### Answer:

The equation of the given curve is *y* = *x*^{3} + 2*x* + 6.

The slope of the tangent to the given curve at any point (*x*, *y*) is given by,

∴ Slope of the normal to the given curve at any point (*x*, *y*) =

The equation of the given line is *x* + 14*y* + 4 = 0.

*x* + 14*y* + 4 = 0 ⇒ (which is of the form *y* = *mx* + *c)*

∴Slope of the given line =

If the normal is parallel to the line, then we must have the slope of the normal being equal to the slope of the line.

When *x* = 2, *y* = 8 + 4 + 6 = 18.

When *x* = −2, *y* = − 8 − 4 + 6 = −6.

Therefore, there are two normals to the given curve with slopeand passing through the points (2, 18) and (−2, −6).

Thus, the equation of the normal through (2, 18) is given by,

And, the equation of the normal through (−2, −6) is given by,

Hence, the equations of the normals to the given curve (which are parallel to the given line) are

#### Question 22:

Find the equations of the tangent and normal to the parabola *y*^{2} = 4*ax* at the point (*at*^{2}, 2*at*).

#### Answer:

The equation of the given parabola is *y*^{2} = 4*ax*.

On differentiating *y*^{2} = 4*ax* with respect to *x*, we have:

∴The slope of the tangent atis

Then, the equation of the tangent atis given by,

*y* − 2*at* =

Now, the slope of the normal atis given by,

Thus, the equation of the normal at (*at*^{2}, 2*at*) is given as:

#### Question 23:

Prove that the curves *x* = *y*^{2} and *xy = k* cut at right angles if 8*k*^{2} = 1. [**Hint**: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]

#### Answer:

The equations of the given curves are given as

Putting *x* = *y*^{2} in *xy* = *k*, we get:

Thus, the point of intersection of the given curves is.

Differentiating *x* = *y*^{2} with respect to *x*, we have:

Therefore, the slope of the tangent to the curve *x* = *y*^{2 }atis

On differentiating *xy* = *k* with respect to *x*, we have:

∴ Slope of the tangent to the curve *xy = k* atis given by,

We know that two curves intersect at right angles if the tangents to the curves at the point of intersection i.e., at are perpendicular to each other.

This implies that we should have the product of the tangents as − 1.

Thus, the given two curves cut at right angles if the product of the slopes of their respective tangents at is −1.

Hence, the given two curves cut at right angels if 8*k*^{2} = 1.

#### Question 24:

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

#### Answer:

Differentiatingwith respect to *x*, we have:

Therefore, the slope of the tangent atis .

Then, the equation of the tangent atis given by,

Now, the slope of the normal atis given by,

Hence, the equation of the normal atis given by,

#### Question 25:

Find the equation of the tangent to the curve which is parallel to the line 4*x* − 2*y* + 5 = 0.

#### Answer:

The equation of the given curve is

The slope of the tangent to the given curve at any point (*x*, *y*) is given by,

The equation of the given line is 4*x* − 2*y* + 5 = 0.

4*x* − 2*y* + 5 = 0 ⇒ (which is of the form

∴Slope of the line = 2

Now, the tangent to the given curve is parallel to the line 4*x* − 2*y* − 5 = 0 if the slope of the tangent is equal to the slope of the line.

∴Equation of the tangent passing through the point is given by,

Hence, the equation of the required tangent is.

#### Question 26:

The slope of the normal to the curve *y* = 2*x*^{2} + 3 sin *x* at *x* = 0 is

(A) 3 (B) (C) −3 (D)

#### Answer:

The equation of the given curve is.

Slope of the tangent to the given curve at *x* = 0 is given by,

Hence, the slope of the normal to the given curve at *x* = 0 is

The correct answer is D.

#### Question 27:

The line *y* = *x* + 1 is a tangent to the curve *y*^{2} = 4*x* at the point

(A) (1, 2) (B) (2, 1) (C) (1, −2) (D) (−1, 2)

#### Answer:

The equation of the given curve is.

Differentiating with respect to *x*, we have:

Therefore, the slope of the tangent to the given curve at any point (*x*, *y*) is given by,

The given line is *y* = *x* + 1 (which is of the form *y* = *mx* + *c)*

∴ Slope of the line = 1

The line *y* = *x* + 1 is a tangent to the given curve if the slope of the line is equal to the slope of the tangent. Also, the line must intersect the curve.

Thus, we must have:

Hence, the line *y* = *x* + 1 is a tangent to the given curve at the point (1, 2).

The correct answer is A.