Given : A circle, 2 x 2 + 2 y 2 = 5 and a parabola, y 2 = 4 5 x . Statement - I : An equation of a common tangent to these curves is y = x + 5 . Statement - II : If the line, y = m x + 5 m m ≠ 0 is their common tangent, then m satisfies m 4 - 3 m 2 + 2 = 0 .

Statement - I is true; Statement - II is true; Statement - II is not a correct explanation for statement - I .

Statement - I is true; Statement - II is false.

Statement - I is false; Statement - II is true.

Statement - I is true; Statement - II is true; Statement - II is a correct explanation for statement - I .

For a positive integer k , let P k be the parabola given by y = x 2 k - 5 and let C be the circle given by x 2 + y 2 = 25 . Then

there is a k such that P k and C intersect each other exactly once

there are infinitely many values of k such that P k and C intersect each other at more than one point

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Minimum distance between the curves $y^{2} = x - 1$ and $x^{2} = y - 1$ is equal to

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Important Questions on Parabola

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

If the common tangent to the parabolas,

y^{2} = 4 x a n d x^{2} = 4 y

also touches the circle,

x^{2} + y^{2} = c^{2},

then

c

is equal to :

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

Given : A circle,

2 x^{2} + 2 y^{2} = 5

and a parabola,

y^{2} = 4 \sqrt{5} x

.

Statement - I : An equation of a common tangent to these curves is

y = x + \sqrt{5}

.

Statement - II : If the line,

y = m x + \frac{\sqrt{5}}{m} (m \neq 0)

is their common tangent, then

m

satisfies

m^{4} - 3 m^{2} + 2 = 0

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

If the line

a x + y = c

, touches both the curves

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

and

y^{2} = 4 \sqrt{2} x

, then

|c|

is equal to:

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

The shortest distance between the parabolas

2 y^{2} = 2 x - 1, 2 x^{2} = 2 y - 1

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

If the tangent at

(1, 7)

to the curve

x^{2} = y - 6

touch the circle

x^{2} + y^{2} + 16 x + 12 y + c = 0

then the value of

c

is:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

Number of common tangents of

y = x^{2}

and

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

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

The tangent to the parabola

y^{2} = 4 x

at the point where it intersects the circle

x^{2} + y^{2} = 5

in the first quadrant, passes through the point:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

A tangent line

L

is drawn at the point

(2, - 4)

on the parabola

y^{2} = 8 x .

If the line

L

is also tangent to the circle

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

then

a

is equal to .

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

Let

P

be the point on the parabola,

y^{2} = 8 x

which is at a minimum distance from the center

C

of the circle

x^{2} + {(y + 6)}^{2} = 1

. Then the equation of the circle, passing through

C

and having its center at

P

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

The common tangents to the circle

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

and the parabola

y^{2} = 8 x

touch the circle at the points

P,

Q

and the parabola at the points

R, S

. Then the area of the quadrilateral

S R P Q

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

The equation of common tangent to the parabola

y^{2} = 4 a x

and

x^{2} = 4 a y

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

Let

P

be a point on the parabola

x^{2} = 4 y

. If the distance of

P

from the center of the circle

x^{2} + y^{2} + 6 x + 8 = 0

is minimum, then the equation of the tangent to the parabola at

P

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

A line is a common tangent to the circle

{(x - 3)}^{2} + y^{2} = 9

and the parabola

y^{2} = 4 x

. If the two points of contact

(a, b)

and

(c, d)

are distinct and lie in the first quadrant, then

2 (a + c)

is equal to ___ .

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

Radius of the largest circle which passes through the focus of the parabola

y^{2} = 4 x

and contained in it, is

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

The equation of the common tangent to the parabola

y = x^{2}

and

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

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

For a positive integer $k$ , let $P_{k}$ be the parabola given by $y = \frac{x^{2}}{k} - 5$ and let $C$ be the circle given by $x^{2} + y^{2} = 25$ .

Then

EASY

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

A straight line is normal to both the parabolas

y^{2} = x

and

x^{2} = y

. The distance of the origin from it is

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

Let

P

and

Q

be distinct points on the parabola

y^{2} = 2 x

such that a circle with

P Q

as diameter passes through the vertex

O

of the parabola. If

P

lies in the first quadrant and the area of the triangle

Δ O P Q

3 \sqrt{2}

sq. units, then which of the following is (are) the coordinates of

P

HARD

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

If the common tangents to the parabola,

x^{2} = 4 y

and the circle,

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

intersect at the point

P

, then the distance of

P

from the origin (units), is:

EASY

Mathematics>Coordinate Geometry>Parabola>Interaction Between Two Curves

If the parabolas

y^{2} = 4 b (x - c)

and

y^{2} = 8 a x

have a common normal, then which one of the following is a valid choice for the ordered triad

(a, b, c)

Minimum distance between the curves y2=x-1 and x2=y-1 is equal to

Important Questions on Parabola

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Minimum distance between the curves $y^{2} = x - 1$ and $x^{2} = y - 1$ is equal to