In a rectangular coordinate plane, AB is the diameter of a circle and point C lies on the circle. If the coordinates of points A and B are (-1,0) and (5,0), and the area of triangle ABC is 6andradic;2 square units, which of the following can be the coordinates of point C?a)b)c)d)e)Correct answer is option 'B'. Can you explain this answer?

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In a rectangular coordinate plane, AB is the diameter of a circle and point C lies on the circle. If the coordinates of points A and B are (-1,0) and (5,0), and the area of triangle ABC is 6√2 square units, which of the following can be the coordinates of point C?

Correct answer is option 'B'. Can you explain this answer?

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In a rectangular coordinate plane, AB is the diameter of a circle and ...

Given:

The given information corresponds to the following diagram:

In ΔABC

Length of side AB = 5 – (-1) = 6 units
Side AB is the diameter of the circle
- So, ∠ACB is the angle in a semicircle
- Therefore, ∠ABC = 90⁰
- Also, Radius of circle = 6/2 = 3 units
- So, OB = 3 units, where O is the center of the circle
- The coordinate of point O will be (2,0)

Area of ΔABC = 6√2 square units

To find: Possible coordinates of point C

Approach and Working:

We know the area of ΔABC.
As per the formula, the area of this triangle = ½ * base * height = ½ * AB * the perpendicular dropped from C on AB.
Now AB lies on the x-axis. So, what will be the length of the perpendicular dropped from C on AB ?
- It will be equal to the y-coordinate of C.

So, we can write:
- ½ (AB)(y-coordinate of point C) = 6√2
- ½* 6* (y-coordinate of point C) = 6√2
- (y-coordinate of point C) = 2√2
To find the x-coordinate of C, we can use the distance formula.
The distance between the center and C is 3 units. Thus, we can write:
- D = √[(x-2)² + (2√2 -0)²]
- 3 = √[(x²-4x +4 + 8]
- x² -4x +12 = 9
- x² -4x + 3 = 0
- (x-1)* (x-3) = 0
- Therefore, x can be either 1 or 3.

Looking at the answer choices, we see that only option B (1,2√2) satisfy.
Thus, the correct answer is Option B.

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