three sides of a triangle have the equations lr equiv y mrx

Three sides of a triangle have the equations $${L_r} \equiv y - {m_r}x - {c_r} = 0;\,r = 1,\,2,\,3.$$ Then $$\lambda {L_2}{L_3} + \mu {L_3}{L_1} + \nu {L_1}{L_2} = 0,$$ where $$\lambda \ne 0,\,\mu \ne 0,\,\nu \ne 0,$$ is the equation of the circumcircle of the triangle if :

A. $$\lambda \left( {{m_2} + {m_3}} \right) + \mu \left( {{m_3} + {m_1}} \right) + \nu \left( {{m_1} + {m_2}} \right) = 0$$

B. $$\lambda \left( {{m_2}{m_3} - 1} \right) + \mu \left( {{m_3}{m_1} - 1} \right) + \nu \left( {{m_1}{m_2} - 1} \right) = 0$$

C. both (A) and (B) hold together

D. none of these

Answer : both (A) and (B) hold together

Solution :
$$\lambda {L_2}{L_3} + \mu {L_3}{L_1} + \nu {L_1}{L_2} = 0$$ is a curve passing through the vertices, which are obtained by solving any two of $${L_1} = 0,\,{L_2} = 0,\,{L_3} = 0$$ together.
It will be the circumcircle if the coefficient of $${x^2} = $$ the coefficient of $${y^2}$$ and the coefficient of $$xy=0.$$

Releted MCQ Question on
Geometry >> Circle

Releted Question 1

A square is inscribed in the circle $${x^2} + {y^2} - 2x + 4y + 3 = 0.$$ Its sides are parallel to the coordinate axes. The one vertex of the square is-

A. $$\left( {1 + \sqrt 2 ,\, - 2 } \right)$$

B. $$\left( {1 - \sqrt 2 ,\, - 2 } \right)$$

C. $$\left( {1 - 2 ,\, + \sqrt 2 } \right)$$

D. none of these

View Answer

Releted Question 2

Two circles $${x^2} + {y^2} = 6$$ and $${x^2} + {y^2} - 6x + 8 = 0$$ are given. Then the equation of the circle through their points of intersection and the point $$\left( {1,\,1} \right)$$ is-

A. $${x^2} + {y^2} - 6x + 4 = 0$$

B. $${x^2} + {y^2} - 3x + 1 = 0$$

C. $${x^2} + {y^2} - 4y + 2 = 0$$

D. none of these

View Answer

Releted Question 3

The centre of the circle passing through the point (0, 1) and touching the curve $$y = {x^2}$$ at $$\left( {2,\,4} \right)$$ is-

A. $$\left( {\frac{{ - 16}}{5},\,\frac{{27}}{{10}}} \right)$$

B. $$\left( {\frac{{ - 16}}{7},\,\frac{{53}}{{10}}} \right)$$

C. $$\left( {\frac{{ - 16}}{5},\,\frac{{53}}{{10}}} \right)$$

D. none of these

View Answer

Releted Question 4

The equation of the circle passing through $$\left( {1,\,1} \right)$$ and the points of intersection of $${x^2} + {y^2} + 13x - 3y = 0$$ and $$2{x^2} + 2{y^2} + 4x - 7y - 25 = 0$$ is-

A. $$4{x^2} + 4{y^2} - 30x - 10y - 25 = 0$$

B. $$4{x^2} + 4{y^2} + 30x - 13y - 25 = 0$$

C. $$4{x^2} + 4{y^2} - 17x - 10y + 25 = 0$$

D. none of these

View Answer

Three sides of a triangle have the equations $${L_r} \equiv y - {m_r}x - {c_r} = 0;\,r = 1,\,2,\,3.$$ Then $$\lambda {L_2}{L_3} + \mu {L_3}{L_1} + \nu {L_1}{L_2} = 0,$$ where $$\lambda \ne 0,\,\mu \ne 0,\,\nu \ne 0,$$ is the equation of the circumcircle of the triangle if :

Releted MCQ Question on
Geometry >> Circle

A square is inscribed in the circle $${x^2} + {y^2} - 2x + 4y + 3 = 0.$$ Its sides are parallel to the coordinate axes. The one vertex of the square is-

Two circles $${x^2} + {y^2} = 6$$ and $${x^2} + {y^2} - 6x + 8 = 0$$ are given. Then the equation of the circle through their points of intersection and the point $$\left( {1,\,1} \right)$$ is-

The centre of the circle passing through the point (0, 1) and touching the curve $$y = {x^2}$$ at $$\left( {2,\,4} \right)$$ is-

The equation of the circle passing through $$\left( {1,\,1} \right)$$ and the points of intersection of $${x^2} + {y^2} + 13x - 3y = 0$$ and $$2{x^2} + 2{y^2} + 4x - 7y - 25 = 0$$ is-

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Circle

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Three sides of a triangle have the equations $${L_r} \equiv y - {m_r}x - {c_r} = 0;\,r = 1,\,2,\,3.$$ Then $$\lambda {L_2}{L_3} + \mu {L_3}{L_1} + \nu {L_1}{L_2} = 0,$$ where $$\lambda \ne 0,\,\mu \ne 0,\,\nu \ne 0,$$ is the equation of the circumcircle of the triangle if :

Releted MCQ Question on Geometry >> Circle

A square is inscribed in the circle $${x^2} + {y^2} - 2x + 4y + 3 = 0.$$ Its sides are parallel to the coordinate axes. The one vertex of the square is-

Two circles $${x^2} + {y^2} = 6$$ and $${x^2} + {y^2} - 6x + 8 = 0$$ are given. Then the equation of the circle through their points of intersection and the point $$\left( {1,\,1} \right)$$ is-

The centre of the circle passing through the point (0, 1) and touching the curve $$y = {x^2}$$ at $$\left( {2,\,4} \right)$$ is-

The equation of the circle passing through $$\left( {1,\,1} \right)$$ and the points of intersection of $${x^2} + {y^2} + 13x - 3y = 0$$ and $$2{x^2} + 2{y^2} + 4x - 7y - 25 = 0$$ is-

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Releted MCQ Question on
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