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Number of Common Tangents to Two Circles in Different Conditions

Let the circl...

Question

Let the circles $C_{1} : x^{2} + y^{2} = 9$ and $C_{2} : (x - 3)^{2} + (y - 4)^{2} = 16,$ intersect at the points $X$ and $Y .$ Suppose that another circle $C_{3} : (x - h)^{2} + (y - k)^{2} = r^{2}$ satisfies the following conditions:

$(i)$ centre of $C_{3}$ is collinear with the centres of $C_{1}$ and $C_{2} .$

$(i i)$ $C_{1}$ and $C_{2}$ both lie inside $C_{3}$ , and

$(i i i)$ $C_{3}$ touches $C_{1}$ at $M$ and $C_{2}$ at $N$

Let the line through $X$ and $Y$ intersect $C_{3}$ at $Z$ and $W$ , and let a common tangent of $C_{1}$ and $C_{3}$ be the tangent to the parabola $x^{2} = 8 α y .$
There are some expressions given in $L i s t - I$ whose values are given in $L i s t - I I$ below:

$\begin{matrix} List I & List I I (I) & 2 h + k & (P) & 6 (I I) & \frac{length of Z W}{length of X Y} & (Q) & \sqrt{6} (I I I) & \frac{Area of triangle M Z N}{Area of triangle Z M W} & (R) & \frac{5}{4} (I V) & α & (S) & \frac{21}{5} (T) & 2 \sqrt{6} (U) & \frac{10}{3} \end{matrix}$

Which of the following is the only INCORRECT combination?

A

(I), (P)

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B

(I V), (U)

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C

(I I I), (R)

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D

(I V), (S)

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Solution

The correct option is D $(I V), (S)$

Let length of perpendicular from $M$ to $Z W$ be $λ$
$λ = 3 + \frac{9}{5} = \frac{24}{5}$
$∴ \frac{Area of triangle M Z N}{Area of triangle Z M W} = \frac{\frac{1}{2} (M N) \frac{1}{2} (Z W)}{\frac{1}{2} (Z W) λ} = \frac{5}{4}$
$C_{3} : {(x - \frac{9}{5})}^{2} + {(y - \frac{12}{5})}^{2} = 6^{2}$
$C_{1} : x^{2} + y^{2} = 9$
Common tangent to $C_{1}$ and $C_{3}$ is $C_{1} - C_{3} = 0 \Rightarrow 3 x + 4 y + 15 = 0$
Now, $3 x + 4 y + 15 = 0$ is tangent to parabola
$x^{2} = 8 α y \Rightarrow x^{2} = 8 α (\frac{- 3 x - 15}{4})$
$\Rightarrow x^{2} + 6 α x + 30 α = 0$
$∴ D = 0 = b^{2} - 4 a c$
$\Rightarrow α = \frac{10}{3}$
So, $α = \frac{21}{5}$ is an incorrect combination.

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Similar questions

Q. Let the circles

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

C_{2} : (x - 3)^{2} + (y - 4)^{2} = 16,

intersect at the points

X

Y .

Suppose that another circle

C_{3} : (x - h)^{2} + (y - k)^{2} = r^{2}

satisfies the following conditions:

(i)

C_{3}

is collinear with the centres of

C_{1}

C_{2} .

(i i)

C_{1}

C_{2}

both lie inside

C_{3}

(i i i)

C_{3}

C_{1}

M

C_{2}

N

Let the line through

X

Y

C_{3}

Z

W

, and let a common tangent of

C_{1}

C_{3}

be the tangent to the parabola

x^{2} = 8 α y .

There are some expressions given in

L i s t - I

whose values are given in

L i s t - I I

\begin{matrix} List I & List I I (I) & 2 h + k & (P) & 6 (I I) & \frac{length of Z W}{length of X Y} & (Q) & \sqrt{6} (I I I) & \frac{Area of triangle M Z N}{Area of triangle Z M W} & (R) & \frac{5}{4} (I V) & α & (S) & \frac{21}{5} (T) & 2 \sqrt{6} (U) & \frac{10}{3} \end{matrix}

Which of the following is the only CORRECT combination?

Q. Let the circles

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

C_{2} : (x - 3)^{2} + (y - 4)^{2} = 16,

intersect at the points

X

Y .

Suppose that another circle

C_{3} : (x - h)^{2} + (y - k)^{2} = r^{2}

satisfies the following conditions:

(i)

C_{3}

is collinear with the centres of

C_{1}

C_{2} .

(i i)

C_{1}

C_{2}

both lie inside

C_{3}

(i i i)

C_{3}

C_{1}

M

C_{2}

N

Let the line through

X

Y

C_{3}

Z

W

, and let a common tangent of

C_{1}

C_{3}

be the tangent to the parabola

x^{2} = 8 α y .

There are some expressions given in

L i s t - I

whose values are given in

L i s t - I I

\begin{matrix} List I & List I I (I) & 2 h + k & (P) & 6 (I I) & \frac{length of Z W}{length of X Y} & (Q) & \sqrt{6} (I I I) & \frac{Area of triangle M Z N}{Area of triangle Z M W} & (R) & \frac{5}{4} (I V) & α & (S) & \frac{21}{5} (T) & 2 \sqrt{6} (U) & \frac{10}{3} \end{matrix}

Which of the following is the only INCORRECT combination?

C_{1} : x^{2} + y^{2} = 3

, with centre at

O

, intersect the parabola

x^{2} = 2 y

P

in the first quadrant. Let the tangent to the circle

C_{1}

P

touches other two circle

C_{2}

C_{3}

R_{2}

R_{3}

, respectively. Suppose

C_{2}

C_{3}

have equal radii

2 \sqrt{3}

Q_{2}

Q_{3}

, respectively. If

Q_{2}

Q_{3}

y

2 x = y

intersects the circle

C_{1} : x^{2} + y^{2} = 5

at P in the 1st quadrant.

C_{2}

C_{3}

are circles, both with radii equal to

2 \sqrt{5}

with respective centers

Q_{2}

Q_{3}

lying on the y-axis. If the tangent to

C_{1}

P

touches the circles

C_{2}

C_{3}

then the distance

Q_{2} Q_{3}

Let $c_{1} : x^{2} + y^{2} = 1; C_{2} : (x - 10)^{2} + y^{2} = 1 a n d C_{3}; x^{2} + y^{2} - 10 x - 42 y + 457 = 0$ be three circle.A circle C has been drawn to touch circles $C_{1}$ and $C_{2}$ externally and $C_{3}$ internally. Now circles $C_{1}, C_{2}$ and $C_{3}$ start rolling on the circumference of circle C in anticlockwise direction with constant speed. The centroid of the triangle formed by joining the centres of rolling circles $C_{1}, C_{2}$ and $C_{3}$ lies on

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