## Chapter 16 – Tangents and Normals Exercise Ex. 16.1

## Chapter 16 – Tangents and Normals Exercise Ex. 16.2

Find the equations of

the tangent and the normal to the given curves at the indicated points:

Find the equations of the tangent

and the normal to the following curves at the indicated points:

X = 3 cosθ

– cos^{3}θ , y = 3 sinθ

– sin^{3}θ

Find the equation of the tangents to

the curve

3x^{2} – y^{2} = 8, which passes through the point (4/3, 0).

## Chapter 16 – Tangents and Normals Exercise Ex. 16.3

Find the angle of intersection of

the following curves:

Y = 4 – x^{2} and y = x^{2}

Prove that the curves xy =

4 and x^{2} + y^{2} = 8 touch each other.

Prove that the curves y^{2} = 4x and x^{2}

+ y^{2} – 6x + 1 = 0 touch each other at the point (1, 2)

## Chapter 16 – Tangents and Normals Exercise Ex. 16VSAQ

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## Chapter 16 – Tangents and Normals Exercise MCQ

The equation to the normal to the curve y = sin x at (0,0) is

- x = 0
- y = 0
- x + y = 0
- x – y = 0

Correct option: (c)

The equation of the normal to the curve y = x + sin x cos x at x = π/2 is

- x = 2
- x = π
- x + π = 0
- 2x = π

Correct option: (d)

The equation of the normal to the curve y = x (2-x) at the point (2,0) is

- x – 2y = 2
- x – 2y + 2 = 0
- 2x + y = 4
- 2x + y – 4 = 0

Correct option: (a)

The point on the curve y^{2} = x where tangent makes 45° angles with x-axis is

- (1/2,1/4)
- (1/4,1/2)
- (4,2)
- (1,1)

Correct option: (b)

If the tangent to the curve x = at^{2}, y=2at is perpendicular to x-axis , then its point of contact is

- (a, a)
- (0, a)
- (0, 0)
- (a, 0)

Correct option: (c)

The point on the curve y = x^{2} – 3x + 2 where tangent is perpendicular to y = x is

- (0,2)
- (1,0)
- (-1,6)
- (2,-2)

Correct option: (b)

The point on the curve y^{2} = x where tangent makes 45° angle with x-axis is

- (1/2,1/4)
- (1/4,1/2)
- (4,2)
- (1,1)

Correct option:(b)

The point on the curve y = 12x – x^{2} where the slope of the tangent is zero will be

- (0,0)
- (2,16)
- (3,9)
- (6,36)

Correct option: (d)

The angle between the curves y^{2} = x and x^{2} = y at (1,1) is

Correct option: (b)

The equation of the normal to the curve 3x^{2} – y^{2} = 8 which is parallel to x + 3y = 8 is

- x – 3y = 8
- x – 3y + 8 = 0
- x + 3y ± 8 = 0
- x + 3y = 0

Correct option: (c)

The equation of tangent at those points where the curve y = x^{2} – 3x + 2 meets x-axis are

- x – y + 2 = 0 = x – y – 1
- x + y – 1 = 0 = x – y – 2
- x – y – 1 = 0 = x – y
- x – y = 0 = x + y

Correct option: (b)

The slope of tangent to the curve x = t^{2} + 3t – 8, y = 2t^{2} – 2t – 5 at point (2,-1) is

- 22 /7
- 6/7
- -6
- 7/6

Correct option: (b)

The what points the slope of the tangent to the curve x^{2} + y^{2} – 2x – 3 = 0 is zero

- (3,0), (-1,0)
- (3,0), (1,2)
- (-1,0) , (1,2)
- (1,2), (1,-2)

Correct option: (d)

The angle of intersection of the curves xy = a^{2} and x^{2} – y^{2} = 2a^{2} is

- 0°
- 45°
- 90°
- 30°

Correct option: (c)

If the curve ay + x^{2} = 7 and x^{3} = y cut orthogonally at (1,1), then a is equal to

- 1
- -6
- 6
- 0

Correct option: (c)

If the line y = x touches the curve y = x^{2} + bx + c at a point (1,1) then

- b = 1, c = 2
- b =-1, c = 1
- b = 2, c = 1
- b = -2, c = 1

Correct option: (b)

The slope of the tangent to the curve x = 3t^{2} + 1, y = t^{3}-1 at x = 1 is

- 1/2
- 0
- -2
- ∞

Correct option: (b)

The curves y = ae^{x} and y = be^{-x} cut orthogonally, if

- a = b
- a = -b
- ab = 1
- ab =2

Correct option: (c)

The equation of the normal to the curve x = a cos^{3}θ, y = a sin^{3}θ at the point θ = π/4 is

- x = 0
- y = 0
- x = y
- x + y = a

Correct option: (c)

If the curves y = 2 e^{x} and y = ae^{-x} intersect orthogonally, then a =

- 1/2
- -1/2
- 2
- 2e
^{2}

Correct option: (a)

The point on the curve y = 6x – x^{2} at which the tangent to the curve is inclined at π/4 to the line x + y = 0

- (-3,-27)
- (3,9)
- 7/2, 35/4
- (0,0)

Correct option: (b)

The angle of intersection of the parabolas y^{2} = 4 ax and x^{2} = 4ay at the origin is

- π/6
- π/3
- π/2
- π/6

Correct option: (c)

The angle of intersection of the curve y = 2 sin^{2} x and y = cos 2 x at x

- π/4
- π /2
- π /3
- π /6

Correct option: (c)

Any tangent to the curve y = 2x^{7} + 3x + 5

- is parallel to x-axis
- is parallel to y -axis
- makes an acute angle with x- axis
- makes an obtuse angle with x -axis

Correct option: (c)

The point on the curve 9y^{2} = x^{3}, where the normal to the curve makes equal intercepts with the axis is

- (4, ±8/3)
- (-4, 8/3)
- (-4,-8/3)
- (8/3,4)

Correct option: (a)

The slope of the tangent to the curve x = t^{2} + 3t – 8, y = 2t^{2} – 2t – 5 at the point (2,-1) is

- 22/7
- 6/7
- 7/6
- -6/7

Correct option: (b)

The line y = mx + 1 is a tangent to the curve y^{2} = 4x, if the value of m is

- 1
- 2
- 3
- 1/2

Correct option: (a)

The normal at the point (1,1) on the curve 2y + x^{2} = 3 is

- x + y = 0
- x – y = 0
- x + y +1 = 0
- x – y = 1

Correct option: (b)

The normal to the curve x^{2} = 4y passing through (1,2) is

- x + y = 3
- x – y = 3
- x + y = 1
- x – y = 1