Test Index

Circles Part 1

Section: Mathematics

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Question : 84 of 100

Marks: +1, -0

Consider a family of circles which are passing through the point

(-1,1)

and are tangent to

x

(h, k)

are the coordinate of the centre of the circles, then the set of values of

k

is given by the interval

$-\frac{1}{2} \le k \le \frac{1}{2}$
$k \le \frac{1}{2}$
$0 \le k \le \frac{1}{2}$
$k \ge \frac{1}{2}$

Solution:

Equation of circle whose center is

(h, k)

i.e

\;(x-h)^2+(y-k)^2=k^2

(radius of circle

=k

because circle is tangent to

x

-axis)

Equation of circle passing through

(-1,+1)

\; \therefore (-1,-h)^2+(1-k)^2=k^2

\; \Rightarrow 1+h^2+2h+1+k^2-2k=k^2

\; \Rightarrow h^2+2h-2k+2=0

\; D \ge 0

\therefore (2)^2-4 \times 1 \cdot (-2k+2) \ge 0

\Rightarrow 4-4(-2k+2) \ge 0

\Rightarrow 1+2k-2 \ge 0

\Rightarrow k \ge \frac{1}{2}

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