line x = -1 is side BC of equilateral ABC circumscribing circle x^2 + y^2 = a^2

[helpmeplsinmaths]

The line with equation x = -a is the equation of the side BC of an equilateral triangle ABC circumscribing the circle with equation x^2 + y^2 = a^2.
1. Find the equations of AB and AC.
2. Find the equation of the circle circumscribing ABC.

Note: Pls show all working and explain how you got the answer thx

 

[Deleted member 4993]

The line with equation x = -a is the equation of the side BC of an equilateral triangle ABC circumscribing the circle with equation x^2 + y^2 = a^2.
1. Find the equations of AB and AC.
2. Find the equation of the circle circumscribing ABC.

Note: Pls show all working and explain how you got the answer thx

Hint: Draw a sketch of the situation.

x = - a is a vertical line (parallel to y-axis) and is tangent to the given circle whose center is at the origin and radius is = a.

What are your thoughts?

Please share your work with us ...even if you know it is wrong

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[helpmeplsinmaths]

What I did

I'm not even sure where to start btw. I started drawing a diagram with a circle inside a triangle and label top point A, left B and right C. Since we know BC is x=-a, what do I do next to find AB and AC? (For part 1)

 

[helpmeplsinmaths]

Ok, I drew the diagram like you hinted and labelled the line tangent to the circle (x = -a) B at the top and C at the bottom. Then I kinda drew the triangle outside the circle and it looks like I need trigonometry to find AC and AB. Is that right? Since there is a right angle at the x axis, do I use cos for AC and BC?

 

[helpmeplsinmaths]

Answer

Found someone to help me Here's the answer for future references.

We can first determine the side length of ABC.ABC Since equilateral triangle ABCABC circumscribes the circle, they must have the same center - the origin. Let point PP be the x-intercept of the line x=−a.xaThen OP=a.OPa We know that OPOP is a third of the median of triangle ABCABC since it is equilateral. So the median length is just 3a.3a Using the 30-60-90 ratios, the side length of the triangle must be 3‾√a.3a
a) Let BB be above C.C The coordinates for BB are (−a,3√2a).a32a The coordinates for AA are (2a,0)2a0 because OAOA is twice OP.OP The equation of ABAB is y=−3‾√6x+3‾√3.y36x33 By symmetry, the equation of line CBCBis y=36x−3‾√3.y36x33
b) Notice that OAOA is a radius of the circumcircle of triangle ABCABC. We already found that OAOA is 2a.2aSo the equation of the circumcircle of triangle ABCABC is x2+y2=4a2.x2y24a2 Notice that its center is also at the origin.