The two conics b x 2 = y and x 2 a 2 - y 2 b 2 = 1 intersect if a is

⇒ a ∈ [ − 1 2 , 1 2 ] − { 0 }

HARD

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A circle has the same centre as an ellipse and passes through the foci $F_{1} & F_{2}$ of the ellipse, such that the two curves intersect in $4$ points. Let $P$ be any one of their point of intersection. If the major axis of the ellipse is $17$ & the area of the triangle $P F_{1} F_{2}$ is $30$ , then the distance between the foci is -

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Important Questions on Hyperbola

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

Let

T_{1}

and

T_{2}

be two distinct common tangents to the ellipse

E : \frac{x^{2}}{6} + \frac{y^{2}}{3} = 1

and the parabola

P : y^{2} = 12 x

. Suppose that the tangent

T_{1}

touches

P

and

E

at the points

A_{1}

and

A_{2}

, respectively and the tangent

T_{2}

touches

P

and

E

at the points

A_{4}

and

A_{3}

, respectively. Then which of the following statements is(are) true?

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

A common tangent

T

to the curves

C_{1} : \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1

and

C_{2} : \frac{x^{2}}{42} - \frac{y^{2}}{143} = 1

does not pass through the fourth quadrant. If

T

touches

C_{1}

(x_{1}, y_{1})

and

C_{2}

(x_{2}, y_{2})

, then

|2 x_{1} + x_{2}|

is equal to _______.

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

Let

S = \{(x, y) \in ℕ \times ℕ : 9 {(x - 3)}^{2} + 16 {(y - 4)}^{2} \leq 144\}

and

T = \{(x, y) \in ℝ \times ℝ : {(x - 7)}^{2} + {(y - 4)}^{2} \leq 36\}

The

n (S \cap T)

is equal to ______.

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

m

be the slope of common tangent to the circle

x^{2} + y^{2} = 16

and ellipse

\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1

in the first quadrant, then

81 m^{8} =

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

The radius of the circle passing through the point of intersection of ellipse

\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1

and

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

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

The two conics

b x^{2} = y

and

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

intersect if

a

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

Given an ellipse whose equation is

\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1

. A tangent is drawn at any point P on the ellipse. This tangent at the point P intersects the tangents at the extremities of the major axes i.e., x = + 3 and x = -3 at T and T', respectively. Now, taking TT' as diameter, a circle is constructed. For every point P on the ellipse, this circle will always pass through which of the following points ?

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

Let

P

be a variable point on the ellipse

\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1

with foci at

S

and

S^{'} .

A

be the area of triangle

P S S^{'}

, then the maximum value of

A

is :

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

The Minimum distance between the circles

x^{2} + y^{2} = 9

and the curve

2 x^{2} + 10 y^{2} + 6 x y = 1

is :

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

The two conics

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

and

y^{2} = - \frac{b}{a} x

intersect, if

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

If a real circle will pass through the points of intersection of hyperbola

x^{2} - y^{2} = a^{2}

and parabola

y = x^{2}

, then-

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

The equation of common tanget of the curves

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

and

y^{2} = 4 x

are

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

The absolute value of slope of common tangents to parabola

y^{2} = 8 x

and hyperbola

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

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

Maximum number of common normals of

y^{2} = 4 a x

and

x^{2} = 4 b y

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

If the foci of the ellipse

\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1

and the hyperbola

\frac{x^{2}}{4} - \frac{y^{2}}{b^{2}} = 1

coincide, then

b^{2}

is equal to

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

Equation of common tangent between circle

x^{2} + y^{2} = 16

and the ellipse

\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1

in the first quadrant is

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

If the circle

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

intersects the hyperbola

x y = c^{2}

in four points

P (x_{1}, y_{2}), Q (x_{2}, y_{2}), R (x_{3}, y_{3}), S (x_{4}, y_{4}),

then

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

θ_{1}, θ_{2}, θ_{3}, θ_{4}

be eccentric angles of the four concyclic points of the ellipse

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

, then

θ_{1} + θ_{2} + θ_{3} + θ_{4}

(

where

n \in I)

HARD

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

The set of positive values of

' m'

for which a line with slope

' m'

is a common tangent to ellipse

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

and parabola

y^{2} = 4 a x

EASY

Mathematics>Coordinate Geometry>Hyperbola>Interaction between Two Conics

Cotyledons are also called-

Important Questions on Hyperbola

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