Let T be the line passing through the points P - 2 , 7 and Q 2 , - 5 . Let F 1 be the set of all pairs of circles S 1 , S 2 such that T is tangents to S 1 at P and tangent to S 2 at Q , and also such that S 1 and S 2 touch each other at a point, say, M . Let E 1 be the set representing the locus of M as the pair S 1 , S 2 varies in F 1 . Let the set of all straight line segments joining a pair of distinct points of E 1 and passing through the point R 1,1 be F 2 . Let E 2 be the set of the mid-points of the line segments in the set F 2 . Then, which of the following statement(s) is (are) TRUE?

The point 0 , 3 2 does NOT lie in E 1

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Three circles of radii $a, b, c, (a < b < c)$ touch each other externally. If they have $x -$ axis as a common tangent, then:

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Important Questions on Circle

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JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 28

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

EASY

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 29

If a variable line,

3 x + 4 y - λ = 0

is such that the two circles

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

and

x^{2} + y^{2} - 18 x - 2 y + 78 = 0

are on its opposite sides, then the set of all values of

λ

is the interval:

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 30

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

P Q R S

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 31

Match the information given in the three columns of the following table (appropriately).


Column- $1$	Column- $2$	Column- $3$
$(I) x^{2} + y^{2} = a^{2}$	$(i) m y = m^{2} x + a$	$(P) (\frac{a}{m^{2}}, \frac{2 a}{m})$
$(I I) x^{2} + a^{2} y^{2} = a^{2}$	$(i i) y = m x + a \sqrt{m^{2} + 1}$	$(Q) (\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
$(I I I) y^{2} = 4 a x$	$(i i i) y = m x + a \sqrt{m^{2} - 1}$	$(R) (\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
$(I V) x^{2} - a^{2} y^{2} = a^{2}$	$(i v) y = m x + \sqrt{a^{2} m^{2} + 1}$	$(S) (\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

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?

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 32

The equation of the circle passing through the foci of the ellipse

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

having centre at

(0, 3)

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 33

Let

P

be the point on the parabola,

y^{2} = 8 x,

which is at a minimum distance from the centre

C

of the circle,

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

Then, the equation of the circle, passing through

C

and having its centre at

P

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-4>Q 1

Circles are inscribed in the acute angle

α

so that every neighbouring circles touch each other. If the radius of the first circle is

R,

then find the sum of the radii of the first

n

circles in terms of

R

and

α .

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-4>Q 2

Find the equation of the circle passing through the points

A (4, 3)

and

B (2, 5)

and touching the axis of

y

. Also find the point

P

on the

y

-axis such that the angle

A P B

has largest magnitude.

Three circles of radii a, b, c, a<b<c touch each other externally. If they have x- axis as a common tangent, then:

Important Questions on Circle

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Three circles of radii $a, b, c, (a < b < c)$ touch each other externally. If they have $x -$ axis as a common tangent, then: