Question

two circles are drawn through the point (3,2)and (1,4) to touch the x axis .find the rati weo of distance between the centre​

Answer 1

$\sqrt{13} \: is \: to \: \sqrt{17}$

hope this is helpful to you

Answer 2

The Equation of circle is given by:

( x-h)² + (y-k)² = r²..... 1

where (h,k) is the center of the circle and r is the radius of the circle.

The arrangement of Circles is given in the figure.

Solving equation 1, we get,

x² + h² -2hx + y² + k² - 2yk = r²............2

now, as the circles are touching the x axis, r will be equal to the k

r=k

rewriting the equation,

x² + h² -2hx + y² + k² - 2yk =k².......3

putting the value of (x,y) = (1,4) in 3

1+ h² -2h + 16 + k² - 8k - k² = 0

h² -2h - 8k +17 = 0 ................4

Again,

putting the value of (x,y) = (3,2) in 3

3² + h² -2h*3 + 2² + k² - 2*2k -k² = 0

9 + h² -6h + 4 -4k = 0

h² -6h-4k + 13 = 0 .................5

Subtracting equation 4 and 5

h² -2h - 8k +17- (h² -6h-4k + 13)  = 0

h² -2h - 8k +17- h² +6h+4k - 13 =0

4h -4k +4 = 0

4(h-k+1) = 0

h-k+1

Hence,

k= h+1.........................6

putting the value of k in equation 5

h² -6h-4k + 13 = 0

h² -6h-4(h+1) + 13 = 0

h² -6h-4h- 4+13 =0

h² - 10h + 9 = 0

(h-1) (h-9) = 0

h= (1,9)

putting in equation 6

k= 1+1 and k = 9+1

k= (2,10)

The two centers are C1 (1.2) and C2 (9,10)

the ratios the can be calculated as:

For C1 and C2

$\sqrt{(9-1)^{2} + (10-2)^{2} } \\= \sqrt{8^{2} + 8^{2} } \\=\sqrt{64+64$

= 8√2 .......................7

For the cord connecting the 2 circles

$\sqrt{(3-1)^{2} + (2-4)^{2} } \\= \sqrt{2^{2} + 2^{2} } \\=\sqrt{4+4$

= 2√2 .......................8

C1C2/cord connecting the 2 circles= $\frac{8\sqrt{2} }{2\sqrt{2} }$

Ratio is 4:1

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