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If $O T$ and $O N$ are perpendicular dropped from the origin to the tangent and normal to the curve $x = a \sin^{3} t, y = a \cos^{3} t$ at an arbitrary point, then

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Important Questions on Application of Derivatives

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

Let

T

denote a curve

y = y (x)

which is in the first quadrant and let the point

(1, 0)

lie on it. Let the tangent to

T

at a point

P

intersect the

y

-axis at

Y_{P} .

P Y_{P}

has length

1

for each point

P

T,

then which of the following options is/are correct?

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

If the tangent to the curve,

y = x^{3} + a x - b

at the point

(1, - 5)

is perpendicular to the line,

- x + y + 4 = 0,

then which one of the following points lies on the curve?

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

Let

S

be the set of all values of

x

for which the tangent to the curve

y = f (x) = x^{3} - x^{2} - 2 x

(x, y)

is parallel to the line segment joining the points

(1, f (1))

and

(- 1, f (- 1)),

then

S

is equal to

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

The equation of the tangent to

y = - 2 x^{2} + 3

x = 1

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

The point on the curve

y = \sqrt{x - 1}

where the tangent is perpendicular to the line

2 x + y - 5 = 0

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

The equation of a common tangent to the curves,

y^{2} = 16 x

and

x y = - 4,

is:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

If the tangents on the ellipse

4 x^{2} + y^{2} = 8

at the points

(1, 2)

and

(a, b)

are perpendicular to each other, then

a^{2}

is equal to

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

If the lines

x + y = a

and

x - y = b

touch the curve

y = x^{2} - 3 x + 2

at the points where the curve intersects the

x -

axis, then

\frac{a}{b}

is equal to

\dots

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

The tangent at the point

(2, - 2)

to the curve,

x^{2} y^{2} - 2 x = 4 (1 - y)

does not pass through the point:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

Let C be a curve given by

y (x) = 1 + \sqrt{4 x - 3}

x > \frac{3}{4}

. If

P

is a point on C, such that the tangent at

P

has slope

\frac{2}{3}

, then a point through which the normal at

P

passes, is :

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

If the tangent to the curve

y = e^{x}

at a point

(c, e^{c})

and the normal to the parabola

y^{2} = 4 x

at the point

(1, 2)

intersect at the same point on the

x -

axis, then the value of

c

is .....

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

The equation of the tangent and normal to the ellipse

x^{2} + 2 y^{2} + 2 x - 4 y - 14 = 0

(2, - 1)

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

If the tangent to the curve

y = x + \sin y

at a point

(a, b)

is parallel to the line joining

(0, \frac{3}{2})

and

(\frac{1}{2}, 2)

, then

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

If the tangent to the curve

y = \frac{x}{x^{2} - 3}, x \in R, (x \neq \pm \sqrt{3}),

at a point

(α, β) \neq (0, 0)

on it is parallel to the line

2 x + 6 y - 11 = 0,

then:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

Which of the following points lies on the tangent to the curve

x^{4} e^{y} + 2 \sqrt{y + 1} = 3

at the point

(1, 0)

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

For non-zero real numbers

l, m, n

and

a

, let

f (x) = l x^{3} + m x + n

and

f (a) = f (4 a) .

Then the value

x \in [a, 4 a],

at which the tangent to the curve

y = f (x)

is parallel to the

x

-axis, is

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

The line

y = x + 1

is a tangent to the curve

y^{2} = 4 x

at the point

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

If the tangent at a point

P

, with parameter

t

, on the curve

x = 4 t^{2} + 3, y = 8 t^{3} - 1, t \in R,

meets the curve again at a point

Q

, then the coordinates of

Q

are :

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

Suppose the tangent to the parabola

y = x^{2} + p x + q

(0, 3)

has slope

- 1

. Then

p + q

equals

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangent and Normal

The tangent to the curve,

y = x e^{x^{2}}

passing through the point

(1, e)

also passes through the point:

If OT and ON are perpendicular dropped from the origin to the tangent and normal to the curve x=asin3t,y=acos3t at an arbitrary point, then

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

If $O T$ and $O N$ are perpendicular dropped from the origin to the tangent and normal to the curve $x = a \sin^{3} t, y = a \cos^{3} t$ at an arbitrary point, then