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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 CORRECT combination?

A

(I), (S)

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B

(I), (U)

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C

(I I), (Q)

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D

(I I), (T)

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Solution

The correct option is C $(I I), (Q)$

Centre of $C_{1}, C_{2}, C_{3}$ are collinear and $C_{1}, C_{2}$ both lie inside $C_{3} .$
$M N$ is diameter of $C_{3}$
$\Rightarrow M N = M C_{1} + C_{1} C_{2} + C_{2} N$
$\Rightarrow 2 r = 3 + \sqrt{3^{2} + 4^{2}} + 4 = 12$
$\Rightarrow r = 6$
where, $r$ is the radius of $C_{3}$

Suppose centre of $C_{3}$ be $(h, k) = (0 + r_{4} cos θ, 0 + r_{4} sin θ)$ $∵ r_{4} = C_{1} C_{3} = 3 and tan θ = \frac{4}{3}$
$∴ C_{3} = (\frac{9}{5}, \frac{12}{5}) = (h, k)$
$∴ 2 h + k = 6$
Equation of $Z W$ and $X Y$ is common chord of circle $C_{1} = 0$ and $C_{2} = 0$

Equation of $Z W :$ $C_{1} - C_{2} = 0$
$∴ 3 x + 4 y = 9$
Perpendicular distance from centre of $C_{3} to Z W = ∣ ∣ ∣ ∣ ∣ ∣ \frac{3 \cdot \frac{9}{5} + 4 \cdot \frac{12}{5} - 9}{\sqrt{3^{2} + 4^{2}}} ∣ ∣ ∣ ∣ ∣ ∣ = \frac{6}{5}$
Length of $Z W$ is,
$Z W = 2 \sqrt{r^{2} - (Z W)^{2}} = 2 \sqrt{6^{2} - {(\frac{6}{5})}^{2}} = \frac{24}{5} \sqrt{6}$

Perpendicular distance from centre of $C_{1} to X Y = ∣ ∣ ∣ ∣ \frac{3 \cdot 0 + 4 \cdot 0 - 9}{\sqrt{3^{2} + 4^{2}}} ∣ ∣ ∣ ∣ = \frac{9}{5}$
Length of $X Y = 2 \sqrt{3^{2} - (\frac{9}{5})^{2}} = \frac{24}{5}$
$∴ \frac{Length of Z W}{Length of X Y} = \sqrt{6}$

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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}

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Number of Common Tangents to Two Circles in Different Conditions

Standard XII Mathematics

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