if the point r x y is equidistant from the points p a b a b and q b a a b then find distance of p from origin mid point of pq and also prove that xa yb

If the point R(x, y) is equidistant from the points P (a+ b, a-b) and Q (b- a, a+b), then find distance of P from origin, mid-point of PQ and also prove that xa = yb

please explain

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Sneha

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Saturday, 28 November 2020 08:31 AM

Let P(x,y), Q(a+b,b-a) and R(a-b,a+b) be the given points. Then,PQ=PR
$? + = +$
$? {x - (a + b)}^{2} + {y - (b - a)}^{2} = {x - (a - b)}^{2} + {y - (a + b)}^{2}$
$? x^{2} - 2 x (a + b) + (a + b)^{2} + y^{2} - 2 y (b - a) + (b - a)^{2}$
$= x^{2} + (a - b)^{2} - 2 x (a - b) + y^{2} - 2 y (a + b) + (a + b)^{2}$
$? - 2 x (a + b) - 2 y (b - a) = - 2 x (a - b) - 2 y (a + b)$
$? a x + b x + b y - a y = a x - b x + a y + b y$
$? 2 b x = 2 a y ? b x = a y$

Let P(x,y), Q(a+b,b-a) and R(a-b,a+b) be the given points. Then,PQ=PR?+=+?{x-(a+b)}2+{y-(b-a)}2={x-(a-b)}2+{y-(a+b)}2?x2-2x(a+b)+(a+b)2+y2-2y(b-a)+(b-a)2=x2+(a-b)2-2x(a-b)+y2-2y(a+b)+(a+b)2?-2x(a+b)-2y(b-a)=-2x(a-b)-2y(a+b)?ax+bx+by-ay=ax-bx+ay+by?2bx=2ay?bx=ay

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