Fun on circles
On a circle, , two points are taken which makes a line parallel to -axis. Tangents drawn to it are perpendicular to each other. So what would be the locus of these points other than a circle?
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1 solution
On a circle,
x 2 + y 2 = 2 5 ,
Let the one point on a line parallel to X-axis, be (x’,y’) so other point will be (-x’,y’).
So eqn of tangents through these points will be , xx’+yy’=25 and yy’–xx’=25
They both will intersect at point (0,25/y').
They both are perpendicular to each other, so m1*m2=-1
So, (-x')/y' * x'/y' =-1.
So x 2 = y 2 = , this is one locus of points, which can also be written as , x=±y
Also, tangents are perpendicular to each other and it is bisected by Y-axis,
so right angled isosceles triangle is formed by (-x’,y’) , (x’,y’) and (0,25/y').
So 25/y'-y'=x'
Which leads to, y 2 + x y = 2 5 .