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Artist 52 [7]
3 years ago
15

4. Find the center and the radius of the circle which circumscribes the triangle with vertices ai, a, a3. Express the result in

symmetric form.
Mathematics
1 answer:
Mama L [17]3 years ago
3 0
<h2>Answer:</h2>

<u>\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right]</u>

<h2>Step-by-step explanation:</h2>

In the question,

We have to find out the circumcentre of the circle passing through the triangle with the vertices (a₁, b₁), (a₂, b₂) and (a₃, c₃).

So,

The circle is passing through these points the equation of the circle is given by,

x^{2}+y^{2}+ax+by+c=0

On putting the points in the circle we get,

x^{2}+y^{2}+ax+by+c=0\\(a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\and,\\(a_{2})^{2}+(b_{2})^{2}+a(a_{2})+b(b_{2})+c=0\\and,\\(a_{3})^{3}+(b_{3})^{3}+a(a_{3})+b(b_{3})+c=0\\

So,

(a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\a(a_{1})+b(b_{1})+c=-(a_{1})^{2}-(b_{1})^{2}\,.........(1)\\and,\\a(a_{2})+b(b_{2})+c=-(a_{2})^{2}-(b_{2})^{2}\,.........(2)\\and,\\a(a_{3})+b(b_{3})+c=-(a_{3})^{3}-(b_{3})^{3}\,.........(3)\\

On solving these equation using, <u>Matrix method we can get the required equation of the circle,</u>

<u>\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right]</u>

<em><u>This is the required answer.</u></em>

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