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For how many values of p, the circle2 x^2+2 y^2+2 x+4 y-p=0 and the coordinate axes have exactly three common points?

 

Option: 1

0


Option: 2

1


Option: 3

finitely many but more than 1

 


Option: 4

infinitely many


Answers (1)

When the circle intersects the x-axis, y=0

\begin{aligned} & \quad 2 x^2+2(0)^2+2 x+4(0)-p=0 \\ & 2 x^2+2 x-p=0 \\ & x=(-2 \pm \sqrt{(} 4+8 p)) / 4 \end{aligned}

When the circle intersects the y-axis,x=0

\begin{aligned} & \quad 2(0)^2+2 y^2+2(0)+4 y-p=0 \\ & 2 y^2+4 y-p=0 \\ & y=(-2 \pm \sqrt{(1+2 p)) / 2} \end{aligned}

The circle to have exactly three common points with each axis, 

We need to have one intersection point at the origin (0,0), and two other intersection points that are not on the same axis.

the discriminant of the quadratic formula for x to be positive:


\begin{aligned} & 4+8 p>0 \\ & p>-1 / 2 \end{aligned}

the two intersection points with the y-axis to be distinct, we need the discriminant of the quadratic formula for y to be positive:


\begin{aligned} 1+2 p>0 \\ p>-1 / 2 \end{aligned}

So, for the circle to have exactly three common points with each axis,


\begin{aligned} 1+2 p>0 \\ p>-1 / 2 \end{aligned}

Therefore, there are infinitely many values of p for which the circle has exactly three common points with each axis.

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Kshitij

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