The length of the diameter of the circle which touches the X-axis at the point (1,0) and passes through the point (2,3) is

Solution :

Let us assume that the coordinates of the center of the circle are C(h,k) and its radius is r.

Now, since the circle touches X-axis at (1,0), hence its radius should be equal to ordinate of center.

$$\implies$$ r = k

Hence, the equation of circle is $$(x – h)^2 + (y – k)^2$$ = $$k^2$$

Also, given that the circle passes through points (1, 0) and (2, 3). Hence, substituting them, in the equation of circle we get

$$(1 – h)^2 + (0 – k)^2$$ = $$k^2$$       ……(i)

$$(2 – h)^2 + (3 – k)^2$$ = $$k^2$$        ……(ii)

from equations (i) and (ii), we get

k = $$5\over 3$$

Hence, The diameter of the circle is 2k = $$10\over 3$$

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