A variable circle is described to pass through (1,0) and tangent to the curve Y = tan(tan inverse X) the - Maths - Conic Sections

Question

A variable circle is described to pass through (1,0) and tangent to
the curve Y = tan(tan inverse X) the locus of centre of circle is a
parabola whose 1 length of latus rectum is 2 root 2, 2 axis of
symmetry does the equation X + Y = 1, 3 vertex has the coordinates
(3/4, 1/4), 4 directrix is x - y = 0

Aarushi Mishra · Accepted Answer

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Note:tantan-1x=xSo equation of tangent becomes y=xConsider such a circle with center C(h,k)
 Let point of contact of this circle with line y=x be PNote: For all such circle
 CP=SPWe know , by definitionLocus of any point such that its distances from a fixed point and a fixed line are equal is called prabola
 The fixed point is called hte focus and the fixed line is called directrix
Thus focus S=1,0 and equation of directrix is y-x=0We know length of latus rectum=4aAlso distance bewtween focus and directrix=2adistance of y=x from S=ax1+by1+ca2+b2 =1-012+12=122a=124a=22=2Length of latus rectum=2Axis of paraboila is perpendicular to directrix and passes thorugh focusSlope of y=x is m1=1for perpendicualr linesm1m2=-1m2=-1Axis will be a line of slope -1 which passes through 1,0Axis: y-0=-1x-1Axis :y+x-1=0Vertex lies mid way between point of intersection of axis and directrix, and the focusIntersection of directrix and axis
 Put y=x in equation of axisx+x-1=02x=1x=12y=12Midpoint of 12, 12 and 1,0=12+12,12+02=34, 14 Vertex=34, 14