The circle passing through the distinct points (1, t), (t, 1) and (t, t) for all values of t, passes through the point : Kaysons Education

Question

The circle passing through the distinct points (1, t), (t, 1) and (t, t) for all values of t, passes through the point

diffcult

Solution

Correct option is

(1, 1)

Let equation of the circle passing through the given points be

x² + y² + 2gx + 2fy + c = 0 (1)

Then 1 + t² + 2g + 2ft + c = 0 (2)

t² + 1+ 2gt + 2f + c = 0 (3)

and t² + t² + 2gt + 2ft + c = 0 (4)

From (2) and (3), we have

That is, f = g, because t ≠ 1, otherwise the three given points would be the same. Again, from (3) and (4), we get

t² + 1+ 2gt + 2g + c = 0 (5)

and 2t² + 2g t + 2 g t + c = 0 (6)

Subtracting these, we get

Finally, from (6), we get c = –2t² + 2 t(t + 1) = 2t, so that the equation of the circle (1) can be written as

x² + y² – (t + 1)x – (t + 1)y + 2t = 0

which passes through (1, 1) for all values of t, while the points given by (b), (c) and (d) do not satisfy this equation.

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The Circle Passing Through The Distinct Points (1, t), (t, 1) And (t, t) For All Values Of t, Passes Through The Point

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Question

The circle passing through the distinct points (1, t), (t, 1) and (t, t) for all values of t, passes through the point

Solution

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