Points A(1, 0) , B(2, 3) , C(5, 3) and D(6, 0) are concyclic.

Question

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Answer 1

From the figure, it is clear that ABCD is a trapezium, because slopes of $B C$ and $A D$ are same.
Now, by distance formula,
$A B = (3 - 0)^{2} + (2 - 1)^{2} = 10$
And $C D = (3 - 0)^{2} + (5 - 6)^{2} = 10$
$∵ A B = B C$ , $A B C D$ is an isosceles trapezium.
So, $∠ A$ and $∠ D$ are equal.
Therefore, $△ E A D$ and $△ E B C$ are isosceles .
$∴ E A = E D$ and $E B = E C$
$\Rightarrow E A \times E B = E D \times E C$ .
Which satisfies condition of points lying on circle.
Therefore, A, B, C and D are concyclic.
Hence, both the statements are correct and reason is correct explanation of assertion.

Answer 2

From the figure, it is clear that ABCD is a trapezium, because slopes of

B C

and

A D

are same.

Now, by distance formula,

A B = (3 - 0)^{2} + (2 - 1)^{2} = 10

And

C D = (3 - 0)^{2} + (5 - 6)^{2} = 10

∵ A B = B C

,

A B C D

is an isosceles trapezium.

So,

∠ A

and

∠ D

are equal.

Therefore,

△ E A D

and

△ E B C

are isosceles .

∴ E A = E D

and

E B = E C

\Rightarrow E A \times E B = E D \times E C

.

Which satisfies condition of points lying on circle.

Therefore, A, B, C and D are concyclic.

Hence, both the statements are correct and reason is correct explanation of assertion.