An equilateral triangle is inscribed in the circle x2 + y2 = a2 with one of the vertices at (a, 0). What is the equation of the side opposite to this vertex?a)2x – a = 0b)x + a = 0c)2x + a = 0d)3x – 2a = 0Correct answer is option 'C'. Can you explain this answer?

Question

An equilateral triangle is inscribed in the circle x2 + y2 = a2 with one of the vertices at (a, 0). What is the equation of the side opposite to this vertex?a)2x &ndash; a = 0b)x + a = 0c)2x + a = 0d)3x &ndash; 2a = 0Correct answer is option 'C'. Can you explain this answer?

Answer

To find the equation of the side opposite to the vertex (a,0), we need to find the coordinates of the other two vertices of the equilateral triangle.

The given vertex is (a,0). Since the triangle is equilateral, the other two vertices will be symmetrically placed on either side of the y-axis.

Let's call the coordinates of one of the other vertices (x, y). Since the triangle is equilateral, the distance between the vertices (a,0) and (x,y) should be equal to the distance between (a,0) and the other vertex.

So, using the distance formula, we have:
sqrt((x-a)^2 + (y-0)^2) = sqrt((x-(-a))^2 + (y-0)^2)

Simplifying this equation, we get:
(x-a)^2 = (x+a)^2

Expanding both sides, we have:
x^2 - 2ax + a^2 = x^2 + 2ax + a^2

Cancelling out the x^2 and a^2 terms, we get:
-2ax = 2ax

Dividing both sides by 2a, we get:
-x = x

This means that x = 0.

So, one of the other vertices of the equilateral triangle is (0,y).

Now, let's find the value of y.

Since the triangle is equilateral, the distance between the vertices (a,0) and (0,y) should be equal to the radius of the circle, which is a.

Using the distance formula, we have:
sqrt((0-a)^2 + (y-0)^2) = a

Simplifying this equation, we get:
sqrt(a^2 + y^2) = a

Squaring both sides, we have:
a^2 + y^2 = a^2

Cancelling out the a^2 terms, we get:
y^2 = 0

This means that y = 0.

So, the coordinates of the other vertex is (0,0).

Now, let's find the equation of the side opposite to the vertex (a,0) using the two coordinates we found.

The equation of a line passing through two points (x1,y1) and (x2,y2) is given by:
(y-y1) = ((y2-y1)/(x2-x1))(x-x1)

Using the coordinates (a,0) and (0,0), we have:
(y-0) = ((0-0)/(0-a))(x-a)

Simplifying this equation, we get:
y = 0

So, the equation of the side opposite to the vertex (a,0) is y = 0.

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