For the circles C 1- x 2- y 2-2 x - c -0 and C 2- x 2- y 2-2 y - c -0 which of the following is-are correct-A- C 1 touches C 2 if c -1-2B- C 1 cuts orthogonally to C 2 if c -0C- 2-1 lies outside of both the circles if c -6D- Radical axis of circle C 1 and C 2 lies passes through 1-1

The correct options are A C_1 touches C_2 if c=12 nbsp; B C_1 cuts orthogonally to C_2 if c=0 nbsp; C (2,1) lies outside of both the circles if c 6C_1:x^2+ ...

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Byju's Answer

Standard XII

Mathematics

Parametric Equation of a Circle

For the circl...

Question

For the circles $C_{1} : x^{2} + y^{2} + 2 x + c = 0$ and $C_{2} : x^{2} + y^{2} + 2 y + c = 0$ which of the following is/are correct?

C_{1}

touches

C_{2}

c = \frac{1}{2}

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C_{1}

cuts orthogonally to

C_{2}

c = 0

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(2, 1)

lies outside of both the circles if

c > - 6

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Radical axis of circle

C_{1}

and

C_{2}

lies passes through

(1, - 1)

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Solution

The correct options are
A $C_{1}$ touches $C_{2}$ if $c = \frac{1}{2}$
B $C_{1}$ cuts orthogonally to $C_{2}$ if $c = 0$
C $(2, 1)$ lies outside of both the circles if $c > - 6$
$C_{1} : x^{2} + y^{2} + 2 x + c = 0$ and $C_{2} : x^{2} + y^{2} + 2 y + c = 0$
Centre and radius of the circle
$c_{1} = (- 1, 0), r_{1} = \sqrt{1 - c} c_{2} = (0, - 1), r_{2} = \sqrt{1 - c}$

Distance between the centres
$= \sqrt{1^{2} + 1^{2}} = \sqrt{2}$
Sum of radii
$= \sqrt{1 - c} + \sqrt{1 - c} = 2 \sqrt{1 - c}$
Condition for the two circles to touch is,
Distance between centre = Sum of radii
So,
$2 \sqrt{1 - c} = \sqrt{2} \Rightarrow c = \frac{1}{2}$

$C_{1}$ and $C_{2}$ intersects orthognally
$2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2} \Rightarrow 0 + 0 = 2 c \Rightarrow c = 0$

$(2, 1)$ lies outside $C_{1}$
$2^{2} + 1^{2} + 2 \times 2 + c > 0 \Rightarrow c > - 9$
$(2, 1)$ lies outside $C_{2}$
$2^{2} + 1^{2} + 2 \times 1 + c > 0 \Rightarrow c > - 6$
Therefore, when $c > - 6$ the point $(2, 1)$ lies outside of both the circle.

Radical axis of $C_{1}$ and $C_{2}$
$C_{1} - C_{2} = 0 x^{2} + y^{2} + 2 x + c - (x^{2} + y^{2} + 2 y + c) = 0 \Rightarrow 2 x - 2 y = 0$
Hence, $(1, - 1)$ doesn't lies on the radical axis.

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For the circles C1:x2+y2+2x+c=0 and C2:x2+y2+2y+c=0 which of the following is/are correct?

For the circles $C_{1} : x^{2} + y^{2} + 2 x + c = 0$ and $C_{2} : x^{2} + y^{2} + 2 y + c = 0$ which of the following is/are correct?