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  • If  <img alt="\left(m_i, 1 / m_i\right), m_i>0, i=1,2,3,4" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cleft%28m_i%2C%201%20/%20m_i%5Cright%29%2C%20m_i%3E0%2C%20i%3D1%2C2%2C

If  \left(m_i, 1 / m_i\right), m_i>0, i=1,2,3,4 are four distinct points on a circle, what is the value of m_1*m_2*m_3*m_4* ?

Option: 1

0


Option: 2

1


Option: 3

-1


Option: 4

2


Answers (1)

best_answer

We are given that (mi,1/mi) where mi> 0for i =1,2,3,4are four distinct points on a circle.

Let's consider the product of the four values,m_1*m_2*m_3*m_4*

Since the points(mi,1/mi) lie on a circle, we can use a property of circles to find a relationship between the four points.

The property we will use is that for any two points (a, b) and (c, d) on a circle, the product of the distances from each point to a fixed point (the center of the circle) is the same. 

\begin{aligned} & \Rightarrow\left(m_1 * m_2\right) *\left(m_3 * m_4\right)=\left(1 / m_1 * 1 / m_2\right) *\left(1 / m_3 * 1 / m_4\right) \\ \Rightarrow & \left(m_1 * m_2\right) *\left(m_3 * m_4\right)=\left(1 / m_1 * 1 / m_2 * 1 / m_3 * 1 / m_4\right) \\ \Rightarrow & \left(m_1 * m_2\right) *\left(m_3 * m_4\right)=\left(1 / m_1 * m_1\right) *\left(1 / m_2 * m_2\right) *\left(1 / m_3 * m_3\right) *\left(1 / m_4 * m_4\right) \\ \Rightarrow & \left(m_1 * m_2\right) *\left(m_3 * m_4\right)=1 \end{aligned}

Therefore, the value ofm_1*m_2*m_3*m_4*=1

 

Posted by

Suraj Bhandari

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