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  • Which of the following is true for Circles  <img alt="C_1:x^2+y^2-2x+4y=8\\" src="https://entrancecorner.oncodecogs.com/gif.latex?C_1%3Ax%5E2&plus;y%5E2-2x&plus;4y%3D8%5C%5C"

Which of the following is true for Circles  C_1:x^2+y^2-2x+4y=8\\ and C_2 : ax^2+ay^2-2ax+4ay=8\\  where a>0?

Option: 1

C_1 and C_2 are concentric circles for all values of a


Option: 2

C_1 is lying entirely inside C_2, if 0<a<1


Option: 3

C_2 is lying entirely inside C_1 if a>1


Option: 4

All of the above


Answers (1)

\\C_1:x^2+y^2-2x+4y=8 \\ C_2 : x^2+y^2-2x+4y=\frac{8}{a} \\ C_1 \ and\ C_2 \,both\ have\ same\ centre\ (1,-2)

So they are concentric for any value of a

Now radius for first circle = \sqrt{1 + 4 + 8}=\sqrt{13}

Radius for second circle = \sqrt{1 + 4 + \frac{8}{a}}=\sqrt{5 + \frac{8}{a}}

  • For 0 < a < 1

radius of second circle > radius of first circle

As they are concentric, so in this case first circle lies entiely inside second circle

  • For a > 1

radius of second circle < radius of first circle

As they are concentric, so in this case second circle lies entiely inside first circle

So option D is correct

Posted by

Kshitij

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