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Question

A variable circle is described to pass through the point (2,3) and is tangent to the line y=x. The locus of the center of the circle is a conic whose:

A
length of latus rectum is 2
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B
axis of symmetry has the equation x+y=5
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C
vertex has the coordinates (94,114)
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D
distance between focus and (nearest) vertex is 21
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Solution

The correct options are
A length of latus rectum is 2
B axis of symmetry has the equation x+y=5
C vertex has the coordinates (94,114)


Here SP=r=PM
Locus of P is a parabola whose focus is (2,3) and directrix is y=x


distance between focus & directrix AS=2a=|32|2=12
Length of the latus rectum =4a=2
The axis (of symmetry) of the parabola is perpendicular to the directrix and passes through (2,3).
The equation of the axis is x+y=5
The coordinates of the point A(intersection of directrix and axis) is (52,52).The vertex is the midpoint of the line joining A and focus .
Vertex :(94,114)
The distance between focus & vertex is 122unit

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