A point $P$ moves so that the length of the tangent from $P$ to the circle $x^{2} + y^{2} - 2 x - 4 y + 1 = 0$ is three times the distance of $P$ from the point $(1, - 2)$ . Then the locus of $P$ is a straight line. Is this statement true?

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Solution

Let the coordinates of point P is (a,b)
Length of tangent from point P is $\sqrt{(} a^{2} + b^{2} - 2 a - 4 b + 1)$
Length of tangent is equal to three times the distance from point (1,-2)
then locus ,
$\sqrt{(} a^{2} + b^{2} - 2 a - 4 b + 1) = 3 \sqrt{(} a - 1)^{2} + (b + 2)^{2}$
squaring both the sides,
$(a^{2} + b^{2} - 2 a - 4 b + 1) = 9 (a - 1)^{2} + (b + 2)^{2}$
$(a^{2} + b^{2} - 2 a - 4 b + 1) = 9 (a^{2} + 1 - 2 a + b^{2} + 4 + 4 b)$
$8 a^{2} + 8 b^{2} - 16 a + 32 b + 44 = 0$
Hence, the required locus is,
$8 a^{2} + 8 b^{2} - 16 a + 32 b + 44 = 0$
Given statement is false

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A point P moves so that the length of the tangent from P to the circle x2+y2−2x−4y+1=0 is three times the distance of P from the point (1,−2). Then the locus of P is a straight line. Is this statement true?

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