If a point $P$ $P$ has coordinates $(0, - 2)$ $(0,-2)$ and $Q$ $Q$ is any point on the circle, $x^{2} + y^{2} - 5 x - y + 5 = 0$ $x^2+y^2 -5x-y+5=0$ , then what is the maximum value of ${(P Q)}^{2}$ $(PQ)^2$ ?

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If a point $P$ $P$ has coordinates $(0, - 2)$ $(0,-2)$ and $Q$ $Q$ is any point on the circle, $x^{2} + y^{2} - 5 x - y + 5 = 0$ $x^2+y^2 -5x-y+5=0$ , then what is the maximum value of ${(P Q)}^{2}$ $(PQ)^2$ ?

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Answer 1 · 2018-03-18T08:50:38

Answer is (2).

Explanation:

The center of the circle $x^{2} + y^{2} - 5 x - y + 5 = 0$ $x^2+y^2-5x-y+5=0$ is $(\frac{5}{2}, \frac{1}{2})$ $(5/2,1/2)$ and radius is $\sqrt{{(\frac{5}{2})}^{2} + {(\frac{1}{2})}^{2} - 5} = \sqrt{\frac{3}{2}} = \frac{\sqrt{6}}{2}$ $sqrt((5/2)^2+(1/2)^2-5)=sqrt(3/2)=sqrt6/2$

Let us put the values of the corrdinates of $P (0, - 2)$ $P(0,-2)$ in the LHS of the equation of circle $x^{2} + y^{2} - 5 x - y + 5 = 0$ $x^2+y^2-5x-y+5=0$ and we get

$0 + 4 - 0 + 10 + 5 = 19$ $0+4-0+10+5=19$ and as it is is greater than $0$ $0$ , the pint is outside the circle and maximum distance of $P$ $P$ from any pont on the circle would be,

distance of $P$ $P$ from centrer of circle plus radius.

Distance of $P (0, - 2)$ $P(0,-2)$ from center at $(\frac{5}{2}, \frac{1}{2})$ $(5/2,1/2)$ is $\sqrt{{(\frac{5}{2} - 0)}^{2} + {(\frac{1}{2} - (- 2))}^{2}} = \sqrt{\frac{25}{4} + \frac{25}{4}} = \sqrt{\frac{50}{4}} = \frac{5 \sqrt{2}}{2}$ $sqrt((5/2-0)^2+(1/2-(-2))^2)=sqrt(25/4+25/4)=sqrt(50/4)=(5sqrt2)/2$

and hence the maximum value of ${(P Q)}^{2}$ $(PQ)^2$ is

${(\frac{5 \sqrt{2}}{2} + \frac{\sqrt{6}}{2})}^{2}$ $((5sqrt2)/2+sqrt6/2)^2$

= $\frac{{(5 \sqrt{2} + \sqrt{6})}^{2}}{4}$ $(5sqrt2+sqrt6)^2/4$

= $\frac{50 + 6 + 20 \sqrt{3}}{4} = 14 + 5 \sqrt{3}$ $(50+6+20sqrt3)/4=14+5sqrt3$

Hence answer is (2).

graph{(x^2+(y+2)^2-0.01)(x^2+y^2-5x-y+5)=0 [-3.917, 6.083, -2.48, 2.52]}

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If a point $P$ $P$ has coordinates $(0, - 2)$ $(0,-2)$ and $Q$ $Q$ is any point on the circle, $x^{2} + y^{2} - 5 x - y + 5 = 0$ $x^2+y^2 -5x-y+5=0$ , then what is the maximum value of ${(P Q)}^{2}$ $(PQ)^2$ ?

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Explanation:

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If a point $P$ $P$ has coordinates $(0, - 2)$ $(0,-2)$ and $Q$ $Q$ is any point on the circle, $x^{2} + y^{2} - 5 x - y + 5 = 0$ $x^2+y^2 -5x-y+5=0$ , then what is the maximum value of ${(P Q)}^{2}$ $(PQ)^2$ ?