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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=|3−2|√2=1√2 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 12√2unit

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