A region in the x y -plane is bounded by the curve y = 25 - x 2 and the line y = 0 . If the point ( a , a + 1 ) lies in the interior of the region, then

a ∈ ( - ∞ , - 1 ) ∪ ( 3 , ∞ )

EASY

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Let $S \equiv x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ be the equation of the circle, then

Statement I The least and the greatest distance of the point $P (10, 7)$ from the circle $x^{2} + y^{2} - 4 x - 2 y - 20 = 0$ are $5$ and $15 units$ , respectively.

Statement II A point $(x_{1}, y_{1})$ lies outside a circle $S = x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , if $S_{1} > 0$ where $S_{1} = x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c .$

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Important Questions on Circle

EASY

Mathematics>Coordinate Geometry>Circle>Circle and a Point

Let

P (a_{1}, b_{1})

and

Q (a_{2}, b_{2})

be two distinct points on a circle with center

C (\sqrt{2}, \sqrt{3})

. Let

O

be the origin and

O C

be perpendicular to both

C P

and

C Q

. If the area of the triangle

O C P

\frac{\sqrt{35}}{2}

, then

a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}

is equal to __________

HARD

Mathematics>Coordinate Geometry>Circle>Circle and a Point

If a point

P (0, - 2)

and

Q

is any point on the circle,

x^{2} + y^{2} - 5 x - y + 5 = 0

, then the maximum value of

{(P Q)}^{2}

MEDIUM

Mathematics>Coordinate Geometry>Circle>Circle and a Point

The ratio of the largest and shortest distances from the point

(2, - 7)

to the circle

x^{2} + y^{2} - 14 x - 10 y - 151 = 0

EASY

Mathematics>Coordinate Geometry>Circle>Circle and a Point

Let

C_{1}

and

C_{2}

be the centres of the circles

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

and

x^{2} + y^{2} - 6 x - 6 y + 14 = 0

respectively. If

P

and

Q

are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral

P C_{1} Q C_{2}

is :

MEDIUM

Mathematics>Coordinate Geometry>Circle>Circle and a Point

Let the circle

S : 36 x^{2} + 36 y^{2} - 108 x + 120 y + C = 0

be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines,

x - 2 y = 4

and

2 x - y = 5

lies inside the circle

S,

then:

MEDIUM

Mathematics>Coordinate Geometry>Circle>Circle and a Point

If a circle

C

passing through the point

(4, 0)

touches the circle

x^{2} + y^{2} + 4 x - 6 y = 12

externally at the point

(1, - 1),

then the radius of

C

is:

MEDIUM

Mathematics>Coordinate Geometry>Circle>Circle and a Point

The set of values of

k

for which the circle

C : 4 x^{2} + 4 y^{2} - 12 x + 8 y + k = 0

lies inside the fourth quadrant and the point

(1, - \frac{1}{3})

lies on or inside the circle

C

HARD

Mathematics>Coordinate Geometry>Circle>Circle and a Point

If the incentre of an equilateral triangle is

(1, 1)

and the equation of its one side is

3 x + 4 y + 3 = 0

, then the equation of the circumcircle of this triangle is:

HARD

Mathematics>Coordinate Geometry>Circle>Circle and a Point

The point

(- 15, 21)

lies

EASY

Mathematics>Coordinate Geometry>Circle>Circle and a Point

Given the ellipse

(E) 4 x^{2} + 9 y^{2} - 36 = 0,

the circle

(C) x^{2} + y^{2} - 9 = 0

and two points

A (1, 2), B (2, 1),

which of the following is correct?

MEDIUM

Mathematics>Coordinate Geometry>Circle>Circle and a Point

If the point

(1, 4)

lies inside the circle

x^{2} + y^{2} - 6 x + 10 y + p = 0

and the circle does not touch or intersect the coordinate axes, then the set of all possible values of

p

is the interval

MEDIUM

Mathematics>Coordinate Geometry>Circle>Circle and a Point

A region in the

x y

-plane is bounded by the curve

y = \sqrt{25 - x^{2}}

and the line

y = 0

. If the point

(a, a + 1)

lies in the interior of the region, then

HARD

Mathematics>Coordinate Geometry>Circle>Circle and a Point

If the area of an equilateral triangle inscribed in the circle

x^{2} + y^{2} + 10 x + 12 y + c = 0

27 \sqrt{3} sq . units,

then

c

is equal to:

MEDIUM

Mathematics>Coordinate Geometry>Circle>Circle and a Point

Let the point

(p, p + 1)

lie inside the region

E = \{(x, y) : 3 - x \leq y \leq \sqrt{9 - x^{2}}, 0 \leq x \leq 3\} .

If the set of all values of

p

is the interval

(a, b),

then

b^{2} + b - a^{2}

is equal to

________.

MEDIUM

Mathematics>Coordinate Geometry>Circle>Circle and a Point

The circle passing through

(1, - 2)

and touching the axis of

x

(3, 0)

also passes through the point

HARD

Mathematics>Coordinate Geometry>Circle>Circle and a Point

The equation of a circle passing through the point

(1, 1)

and the point of intersection of the circles

x^{2} + y^{2} + 13 x - 3 y = 0

and

2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0

HARD

Mathematics>Coordinate Geometry>Circle>Circle and a Point

If a circle passing through the point

(- 1, 0)

touches

y

-axis at

(0, 2)

, then the

x

-intercept of the circle is

EASY

Mathematics>Coordinate Geometry>Circle>Circle and a Point

(2, a)

does not lie outside the circles

x^{2} + y^{2} = 13

and

x^{2} + y^{2} + x - 2 y = 14,

then

a

lies in

HARD

Mathematics>Coordinate Geometry>Circle>Circle and a Point

The interval of

a

for which the point

(\sqrt{2} a, a + 2)

will be an interior point of the smaller region bounded by the circle

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

and the parabola

x^{2} = 4 y

MEDIUM

Mathematics>Coordinate Geometry>Circle>Circle and a Point

The point

([P + 1], [P])

(where,

[x]

is the greatest integer function) lying inside the region bounded by the circle

x^{2} + y^{2} - 2 x - 15 = 0

and

x^{2} + y^{2} - 2 x - 7 = 0

, then

Important Questions on Circle

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