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  • A square is inscribed in the circle <img alt="2 x^2+2 y^2+6 x+8 y+100=0" src="https://entrancecorner.oncodecogs.com/gif.latex?2%20x%5E2&plus;2%20y%5E2&plus;6%20x&plus;8%20y&plus;100

A square is inscribed in the circle 2 x^2+2 y^2+6 x+8 y+100=0with its sides parallel to the coordinate axes. Then, the distance of the vertex of this square which is nearest to the origin is

 

Option: 1

(-7 / 2,0) \\


Option: 2

(7 / 2,0) \mid \\


Option: 3

(0,-7) \\


Option: 4

(0,7)


Answers (1)

best_answer

The equation of the circle in standard form by completing the square for both x and y terms:aa


\begin{aligned} & \quad 2 x^2+2 y^2+6 x+8 y+100=0 \\ & 2\left(x^2+3 x\right)+2\left(y^2+4 y\right)+100=0 \\ & 2\left(x^2+3 x+9 / 4\right)+2\left(y^2+4 y+4\right)+100=2(9 / 4+4) \\ & 2(x+3 / 2)^2+2(y+2)^2=49 \end{aligned}

Thus, the center of the circle is (-3 / 2,-2)and its radius is\sqrt{(49 / 2)}

the vertices of the inscribed square, we note that they must lie on the circle and have coordinates of the form( \pm r, 0) or (0\pm r, ) where r is the length of a side of the square. 

 Since the sides of the square are parallel to the coordinate axes, we know that the length of each side is the same as the diameter of the circle, which isThus, the vertices of the inscribed square are(-7 / 2,0),(7 / 2,0),(0,-7)and (0,7)

The distance from each vertex to the origin and choose the smallest one:

distance from (-7 / 2,0) to the origin \left.\sqrt{(}-7 / 2)^2\right]=7 / 2

distance from(7,2,3)to the origin \sqrt{(} 7 / 2)^2=7 / 2

distance from(0,-7)to the origin\sqrt{(}-7)^2=7

distance from (0,7)to the origin:\sqrt{(7)^2}=7

Thus, the vertex of the inscribed square nearest to the origin is (-7 / 2,0)and its distance to the origin is7/2

 

 

distance from

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Rishabh

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