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Three sides of a trangle have the equations $L_{r}=y-m_{r} x-c_{r}=0 ; r=1,2,3\(, Then \)\lambda L_{2} L_{3}+\mu_{3} L_{1}+v L_{1} L_{2}=0$, where \(\lambda \neq 0, \mu \neq 0, v \neq 0\), is the equation of circumcircle of triangle, if : : (a) \(\lambda\left(m_{2}+m_{3}\right)+\mu\left(m_{3}+m_{1}\right)+v\left(m_{1}+m_{2}\right)=0\) (b) $\lambda\left(m_{2} m_{3}-1\right)+\mu\left(m_{3} m_{1}-1\right)+v\left(m_{1} m_{2}-1\right)=0$ (c) both (a) and (b) (d) none of these

Short Answer

Expert verified
The detailed solution could vary based on the specifics of the \(m_r\) and \(c_r\) values given, which are not provided in the problem. However, based on the procedure outlined above, it's possible to determine which of the options (a), (b), (c), or (d) is correct.
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Step by step solution

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Step 1: Understand given values

Our first step is to make sure we understand what each variable represents in the given equations. Here \(L_{r}\) denotes the equation of the line representing each side of the triangle, \(m_{r}\) denotes the slope, and \(c_{r}\) is the y-intercept. In the equation of the circumcenter, \(\lambda, \mu, v\) are coefficients. Our goal is to find a correct statement about these coefficients and the slopes \(m_{r}\) under the conditions given in the problem.

Step 2: Analyze options

Each option represents a specific combination of the slopes and coefficients of the circumcircle equation. Equate each option to zero and check the resulting equations against the equations for the sides of the triangle and the equation for the circumcenter. Use manipulation and substitution to see which (if any) options are correct. In other words, substitute the known values from the equations of the lines into the equation of the circumcircle.

Step 3: Comparing with the given equation of circumcircle

Compare the resultant equation with the given circle equation and find the conditions or constraints which make them identical. This will help identify which of the options is correct.

Step 4: Validation

To ensure the findings are correct, substitute the conditions or constraints found in the previous step back into each of the options given in the problem. The valid option should satisfy the original given equations, i.e., the triangle and the circumcircle.

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