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Question

If C is a circle described on the focal chord of the parabola y2=4x as diameter which is inclined at an angle of 45 with the positive xaxis, then

A
Radius of the circle is 2 units
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B
The centre of circle is (3,2)
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C
The line x+1=0 touches the circle
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D
The circle x2+y2+2x6y+3=0 is orthogonal to C
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Solution

The correct option is C The line x+1=0 touches the circle
Given parabola equation is y2=4x
Let P=(t21,2t1),Q=(t21,2t1)
Let PSQ be the focal chord, where S(1,0) is the focus of the parabola
t1t2=1
Since, circle described on focal chord as diameter always touches the directrix,
So, the line x+1=0 always touches the circle.
The slope of PQ is
tan45=12tt21=1t22t1=0
Whose roots are t1,t2, so
t1+t2=2, t1t2=1

The length of focal chord
=4a cosec2θ=4×1×(2)2=8
So, the radius of circle is
r=82=4 units

Now, the equation of the circle described on focal chord as diameter is
(xt21)(xt22)+(y2t1)(y2t2)=0x2(t21+t22)x+t21t22+y2(2t1+2t2)y+4t1t2=0x2+y2(t21+t22)x2(t1+t2)y3=0x2+y2[(t1+t2)22t1t2]x4y3=0x2+y26x4y3=0

Centre (3,2)

Circles x2+y26x4y3=0 and x2+y2+2x6y+3=0 are not orthogonal
2(gg+ff)c+c

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