Question

ABC is a triangle in which angle C is a right angle. If the coordinate A and B are $( - 3,4)$ and $(3, - 4)$ respectively, then the equation of circumcircle of triangle ABC
A ${x^2} + {y^2} - 6x + 8y = 0$
B ${x^2} + {y^2} = 25$
C ${x^2} + {y^2} - 3x + 4y + 5 = 0$
D None of these

Answer 1

Hint: First we will use the equation of circumcircle of triangle then we will put the given points in the place of ${x_1},{x_2},{y_1},{y_2}$. Then we will simplify the resultant equation in order to get the required equation of the circumcircle of triangle ABC.

Formula Used: $(x - {x_1})(x - {x_2}) + (y - {y_1})(y - {y_2}) = 0$

Complete step by step Solution:
Equation of circumcircle of triangle
$(x - {x_1})(x - {x_2}) + (y - {y_1})(y - {y_2}) = 0$ (1)
Where $({x_1},{y_1})$ and $({x_2},{y_2})$ are the vertices of triangle
Here $({x_1},{y_1}) = \left( { - 3,4} \right)$ and $({x_2},{y_2}) = (3, - 4)$
Putting the given points in the equation (1)
$(x + 3)(x - 3) + (y - 4)(y + 4) = 0$
Using the identity $(a - b)(a + b) = ({a^2} - {b^2})$ in the above expression
$({x^2} - {3^2}) + ({y^2} - {4^2}) = 0$
After solving the above equation.
${x^2} - 9 + {y^2} - 16 = 0$
Shifting constant to the other side
${x^2} + {y^2} = 9 + 16$
After further solving
${x^2} + {y^2} = 25$
Hence, the equation of circumcircle of equation is ${x^2} + {y^2} = 25$

Therefore, the correct option is B.

Additional Information: Triangle circumcircle: A triangle's circumcircle is a circle that passes through each of its three vertices. The circumcentre of a triangle is the center of the circle that travels across all of its vertices. When a triangle's vertices are located on a circle, the triangle's sides create the circle's chords.
The separation between a circumcircle's center and any one of its three vertices is known as the radius of the circle. As a result, a triangle's circumcentre is equally spaced from its three vertices. Three distinct lines that are at right angles to the middle of each side of the triangle cross at the center.

Note: Students should know the concept of the circumcircle of the triangle to solve the question. And Should concentrate on the given concept to solve the question without any mistake or difficulty.