Lines CA and CB touch the circle only at points A and B respectively, as shown in the figure above. If the centre O of the circle is at the point (0,0), the coordinates of point C are (0,2) and the radius of the circle is 2 units, what are the coordinates of point A?a)b)c)d)e)Correct answer is option 'C'. Can you explain this answer?

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Lines CA and CB touch the circle only at points A and B respectively, as shown in the figure above. If the centre O of the circle is at the point (0,0), the coordinates of point C are (0,2) and the radius of the circle is √2 units, what are the coordinates of point A?

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

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Lines CA and CB touch the circle only at points A and B respectively, ...

Given:

CA and CB are tangents to the circle
- Since tangents from the same point are equal, CA = CB
- Also, ∠OBC = ∠OAC = 90^o
Center of the circle is O (0,0) and Radius = √2 units
Point C(0,2)

To find:

The x- and y- coordinates of Point A

Approach and Working:

Let’s drop a perpendicular AD from A on the x-axis

So, the x-coordinate of point A = OD
And, the y-coordinate of point A = AD

So, in order to answer the question, we need to find OD and AD
In right ΔAOC, we know sides OA = √2 and OC = 2 units
- Using Pythagoras Theorem, we can write –
  - OC² = OA² + AC²
  - AC²= OC² – OA² = 4 – 2 = 2
  - Hence AC = √2
Thus, we can conclude that ΔAOC is an isosceles right-angled triangle, where angle ACO = AOC = 45^o.
Angle AOD = 90^o – angle AOC = 90⁰ – 45^o = 45^o.

Finding OD and AD
- In triangle AOD, angle AOD = 45, and angle ADO = 90, thus angle DOA = 45
  - Hence Triangle AOD is a 45-45-90 Triangle in which we can write the side ratio as –
    - AD: OD: AO = 1: 1: √2
  - Since the exact value of AO = √2
  - We can conclude from the ratio above, that AD = OD = 1 units
- Thus, we see that the coordinates of point A are (1,1)
Looking at the answer choices, we see that the correct answer is Option C .

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