A point P lies inside the circles $x^{2} + y^{2} - 4 = 0$ and $x^{2} + y^{2} - 8 x + 7 = 0$ . The point P starts moving under the conditions that its path encloses greatest possible area and it is at a fixed distance from any arbitrarily chosen fixed point in its region. The locus of P is

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Solution

$x^{2} + y^{2} = 4$
$x^{2} + y^{2} - 8 x + 7 = 0 \Rightarrow (x - 4)^{2} + y^{2} = (3)^{2}$
According to question point P lies in the region ABCD, and for it to cover maximum distance - the point has to travel along a circle with maximum diameter.
Hence with AB as diameter A(1,0) and B(2,0) - the mid-point is $[\frac{(1 + 2)}{2}, \frac{0}{a}] = (1.5, 0)$ or $(\frac{3}{2}, 0)$ with radius $= \frac{1}{2}$ ; the equation will be
${(x - \frac{3}{2})}^{2} + (y - 0)^{2} = {(\frac{1}{2})}^{2} [∵ (x - a)^{2} + (y - b)^{2} = r^{2}] \Rightarrow x^{2} - 3 x + \frac{9}{4} + y^{2} = \frac{1}{4} \Rightarrow x^{2} + y^{2} - 3 x + 2 = 0$ .

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Similar questions

x^{2} + y^{2} - 4 = 0

x^{2} + y^{2} - 8 x + 7 = 0

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