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 $\sqrt{2}$
• B. axis of symmetry has the equation $x+y=5$
• C. vertex has the coordinates $(\frac{9}{4},\frac{11}{4})$
• D. distance between focus and (nearest) vertex is $\sqrt{2}-1$

Answer 1

Answer: (A), (B), (C)

The correct options are
A length of latus rectum is $\sqrt{2}$
B axis of symmetry has the equation $x+y=5$
C vertex has the coordinates $(\frac{9}{4},\frac{11}{4})$



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


distance between focus & directrix $AS=2a=\frac{|3-2|}{\sqrt{2}}=\frac{1}{\sqrt{2}}$
Length of the latus rectum $=4a=\sqrt{2}$
The axis (of symmetry) of the parabola is perpendicular to the directrix and passes through $(2,3).$
The equation of the axis is $x+y=5$
$\Rightarrow$ The coordinates of the point $A$(intersection of directrix and axis) is $(\frac{5}{2},\frac{5}{2})$.The vertex is the midpoint of the line joining $A$ and focus .
$\Rightarrow$ Vertex $:(\frac{9}{4},\frac{11}{4})$
The distance between focus & vertex is $\frac{1}{2\sqrt{2}}$unit