Two curves a i x 2 + b j y 2 = 1 ; i = 1 , 2 where, a 1 ≠ a 2 , b 1 ≠ b 2 , a 1 , a 2 , b 1 , b 2 ≠ 0 may intersect orthogonally if _________.

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

a 1 - 1 + a 2 - 1 = b 1 - 1 + b 2 - 1

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The acute angle of intersection of the curves $x^{2} y = 1$ and $y = x^{2}$ in the first quadrant is $θ,$ then $\tan θ$ is equal to

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Important Questions on Application of Derivatives

HARD

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

Two curves

a_{i} x^{2} + b_{j} y^{2} = 1; i = 1, 2

where,

a_{1} \neq a_{2},

b_{1} \neq b_{2}, a_{1}, a_{2}, b_{1}, b_{2} \neq 0

may intersect orthogonally if _________.

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

The angle between the curves

x^{2} + y^{2} = 25

and

x^{2} + y^{2} - 2 x + 3 y - 43 = 0

(- 3, 4)

HARD

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

If the curves

x = y^{4}

and

x y = k

cut at right angles, then

{(4 k)}^{6}

is equal to ___ .

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

The angle between the curves

2 x^{2} + y^{2} = 20

and

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

at a point where they intersect in the

4^{th}

quadrant is

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

If the curve

y = a^{x}

and

y = b^{x}

intersect at angle

α

, then

\tan α

is equal to

HARD

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

Let

θ

be the acute angle between the tangents to the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{1} = 1

and the circle

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

at their point of intersection in the first quadrant. Then

\tan θ

is equal to :

HARD

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

P

is a point on the parabola

y = x^{2} + 4

which is closest to the straight line

y = 4 x - 1,

then the co-ordinates of

P

are:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

The curve $y^{2} (x - 2) = x^{2} (1 + x)$ has

EASY

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

If the curves

2 x = y^{2}

and

2 x y = K

intersect perpendicularly, then the value of

K^{2}

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

If the curves

y^{2} = 6 x, 9 x^{2} + b y^{2} = 16

cut each other at right angles, then the value of

b

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

If the curves

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{4} = 1

and

y^{3} = 16 x

intersect at right angles, then

a^{2} =

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

If the curves

y^{2} = 6 x, 9 x^{2} + b y^{2} = 16

intersect each other at right angles, then the value of

b

is:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

If the two curves

x = y^{2}

and

x y = k

cut each other at right angles, then a possible value of

k

HARD

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

An angle of intersection of the curves,

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

and

x^{2} + y^{2} = a b, a > b,

is :

EASY

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

The value of $' a'$ so that the curves $y = 3 e^{x} and y = \frac{a}{3} e^{- x}$ intersect orthogonally is

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

The two curves

x^{3} - 3 x y^{2} + 2 = 0

and

3 x^{2} y - y^{3} - 2 = 0

intersect at an angle of

EASY

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

The value of '

a

' so that the curves

y = 3 e^{x}

and

y = a e^{- x}

intersect orthogonally is

HARD

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

If the curves,

\frac{x^{2}}{a} + \frac{y^{2}}{b} = 1

and

\frac{x^{2}}{c} + \frac{y^{2}}{d} = 1

intersect each other at an angle of

90^{°},

then which of the following relations is TRUE?

HARD

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

Let

P (h, k)

be a point on the curve

y = x^{2} + 7 x + 2

, nearest to the line,

y = 3 x - 3

. Then the equation of the normal to the curve at

P

HARD

Mathematics>Differential Calculus>Application of Derivatives>Interaction between Two Curves

A helicopter is flying along the curve given by

y - x^{\frac{3}{2}} = 7, (x \geq 0) .

A soldier positioned at the point

(\frac{1}{2}, 7)

, who wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:

The acute angle of intersection of the curves x2y=1 and y=x2 in the first quadrant is θ, then tan⁡θ is equal to

Important Questions on Application of Derivatives

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The acute angle of intersection of the curves $x^{2} y = 1$ and $y = x^{2}$ in the first quadrant is $θ,$ then $\tan θ$ is equal to