If 4 a 2 - 5 b 2 + 6 a + 1 = 0 , where a , b ∈ R and the line a x + b y + 1 = 0 touches a fixed circle, then

Center of the circle is - 3 , 0

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If $4 a^{2} - 5 b^{2} + 6 a + 1 = 0$ , where $a, b \in R$ and the line $a x + b y + 1 = 0$ touches a fixed circle, then

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Important Questions on Conic Sections

EASY

Mathematics>Coordinate Geometry>Conic Sections>Circle

The tangent and the normal lines at the point

(\sqrt{3}, 1)

to the circle

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

and the

x

-axis form a triangle. The area of this triangle (in square units) is:

HARD

Mathematics>Coordinate Geometry>Conic Sections>Circle

Let

R S

be the diameter of the circle

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

where,

S

is the point

(1, 0)

. Let

P

be a variable point (other than

R & S

) on the circle and tangents to the circle at

S & P

meet at the point

Q

. The normal to the circle at

P

intersects a line drawn through

Q

parallel to

R S

at point

E

. Then, the locus of

E

passes through the point (s):

HARD

Mathematics>Coordinate Geometry>Conic Sections>Circle

If a tangent to the circle

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

intersects the coordinate axes at distinct points

P

and

Q,

then the locus of the mid-point of

P Q

is:

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>Circle

The equation of the normal to the circle

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

which is parallel to the straight line

x + 2 y = 3

is given by

HARD

Mathematics>Coordinate Geometry>Conic Sections>Circle

A line drawn through the point

P (4, 7)

cuts the circle

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

at the points

A

and

B

. Then

P_{A} \cdot P_{B}

is equal to.

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>Circle

Let the tangents drawn from the origin to the circle,

x^{2} + y^{2} - 8 x - 4 y + 16 = 0

touch it at the points

A

and

B

. Then

{(A B)}^{2}

is equal to

HARD

Mathematics>Coordinate Geometry>Conic Sections>Circle

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

For

a = \sqrt{2}

, if a tangent is drawn to a suitable conic (Column 1) at the point of contact

(- 1, 1)

, then which of the following options is the only Correct combination for obtaining its equation?

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>Circle

y + c = 0

is a tangent to the circle

x^{2} + y^{2} - 6 x - 2 y + 1 = 0

(a, 4)

, then

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>Circle

If the length of the tangent from any point on the circle

{(x - 3)}^{2} + {(y + 2)}^{2} = 5 r^{2}

to the circle

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

16

units, then the area between the two circles in sq. units is

EASY

Mathematics>Coordinate Geometry>Conic Sections>Circle

Equation of the tangent to the circle, at the point

(1, - 1)

, whose center, is the point of intersection of the straight lines

x - y = 1

and

2 x + y = 3

is:

HARD

Mathematics>Coordinate Geometry>Conic Sections>Circle

The diameter of the circle, whose Centre lies on the line

x + y = 2

in the first quadrant and which touches both the lines

x = 3

and

y = 2

EASY

Mathematics>Coordinate Geometry>Conic Sections>Circle

The tangents of two points

A

and

B

on the circle with centre

O

intersect at a point

P .

If in quadrilateral

P A O B,

∠ A O B : ∠ A P B = 5 : 1,

then the measure of

∠ A P B

is given by

HARD

Mathematics>Coordinate Geometry>Conic Sections>Circle

Let

C

be the circle concentric with the circle,

2 x^{2} + 2 y^{2} - 6 x - 10 y = 183

and having area

{(\frac{1}{10})}^{th}

of the area of this circle. Then a tangent to

C,

parallel to the line,

3 x + y = 0

makes an intercept on the

y

-axis, which is equal to

EASY

Mathematics>Coordinate Geometry>Conic Sections>Circle

If a line

y = m x + c

is a tangent to the circle

{(x - 3)}^{2} + y^{2} = 1

and it is perpendicular to a line

L_{1},

where

L_{1}

is the tangent to the circle

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

at the point

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

, then

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>Circle

The angle between the pair of tangents drawn from

(1, 3)

to the circle

x^{2} + y^{2} - 2 x + 4 y - 11 = 0

HARD

Mathematics>Coordinate Geometry>Conic Sections>Circle

P (x_{1}, y_{1})

is a point such that the lengths of the tangents from it to the circles

x^{2} + y^{2} - 4 x - 6 y - 12 = 0

and

x^{2} + y^{2} + 6 x + 18 y + 26 = 0

are in the ratio

2 : 3

, then the locus of

P

HARD

Mathematics>Coordinate Geometry>Conic Sections>Circle

Let

T

be a circle with diameter

A B

and centre

O

. Let

l

be the tangent to

T

B

. For each point

M

T

different from

A

, consider the tangent

t

M

and let interest

l

P .

Draw a line parallel to

A B

through

P

intersecting

O M

Q .

The locus of

Q

M

varies over

T

HARD

Mathematics>Coordinate Geometry>Conic Sections>Circle

Let

E_{1}

and

E_{2}

be two ellipse whose centers are at the origin. The major axes of

E_{1} & E_{2}

lie along the

x

-axis and the

y

-axis, respectively. Let

S

be the circle

x^{2} + {(y - 1)}^{2} = 2 .

The straight line

x + y = 3

touches the curves

S, E_{1}

and

E_{2}

P, Q

and

R,

respectively. Suppose that

P Q = P R = \frac{2 \sqrt{2}}{3}

. If

e_{1}

and

e_{2}

are the eccentricities of

E_{1}

and

E_{2},

respectively, then the correct expression(s) is(are)

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>Circle

The radius of a circle, having minimum area, which touches the curve

y = 4 - x^{2}

and the lines,

y = |x|

is:

MEDIUM

Mathematics>Coordinate Geometry>Conic Sections>Circle

Let the tangents drawn to the circle,

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

from the point

P (0, h)

meet the

x

-axis at points

A

and

B

. If the area of

Δ A P B

is minimum, then positive value of

h

is:

If 4a2-5b2+6a+1=0, where a, b∈R and the line ax+by+1=0 touches a fixed circle, then

Important Questions on Conic Sections

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If $4 a^{2} - 5 b^{2} + 6 a + 1 = 0$ , where $a, b \in R$ and the line $a x + b y + 1 = 0$ touches a fixed circle, then