Consider a family of circles which are passing through the point (–1, 1) and are tangent to x-axis. If  (h,k) are the coordinate of the centre of the circles, then the set of values of k is given by the interval

  • Option 1)

    -\frac{1}{2}\leq k\leq \frac{1}{2}\;

  • Option 2)

    \; k\leq \frac{1}{2}\;

  • Option 3)

    \; 0\leq k\leq \frac{1}{2}\;

  • Option 4)

    \; k\geq \frac{1}{2}

 

Answers (1)

As we learnt in

General form of a circle -

x^{2}+y^{2}+2gx+2fy+c= 0
 

- wherein

centre = \left ( -g,-f \right )

radius = \sqrt{g^{2}+f^{2}-c}

 Equation of circle with center (h,k) is:

(x-h)^{2}+(y-k)^{2}=h^{2}

Radius of circle is  k

(-1-h)^{2}+(1-k)^{2}=k^{2}

1^{2}+h^{2}+2h+1+1+k^{2}-2k=k^{2}

h^{2}+2h-2k+2=0

\Delta \geqslant 0

We get,  4-4(-2k+2) \geqslant 0

k \geqslant \frac{1}{2}


Option 1)

-\frac{1}{2}\leq k\leq \frac{1}{2}\;

This option is incorrect.

Option 2)

\; k\leq \frac{1}{2}\;

This option is incorrect.

Option 3)

\; 0\leq k\leq \frac{1}{2}\;

This option is incorrect.

Option 4)

\; k\geq \frac{1}{2}

This option is correct.

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