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Question

If point (4,4) lies on a parabola whose focus lies on x -axis and directrix is lx+my=1 such that (l,m) lies on curve x2+y2−32xy+8x+8y−1=0, then

A
Co-ordinates of focus may be (5,0)
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B
Co-ordinates of focus may be (3,0)
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C
If tangent at (4,4) passes through origin, value of m may be 13
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D
If tangent at (4,4) passes through origin, equation of axis may be x4y3=0
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Solution

The correct options are
A Co-ordinates of focus may be (5,0)
B Co-ordinates of focus may be (3,0)
C If tangent at (4,4) passes through origin, value of m may be 13
D If tangent at (4,4) passes through origin, equation of axis may be x4y3=0
Let focus of the parabola is (α,0)
Now, SP=PM
(4α)2+(40)2=|4l+4m1|l2+m2



(l2+m2)((α4)2+16)=(4(l+m)1)2
(l2+m2)(α2+328α)=16(l+m)2+18(l+m)
(l2+m2)(α2+328a16)+8(l+m)32lm=1
(l2+m2)(α4)2+8(l+m)32lm1=0

But given that (l2+m2)+8(l+m)32lm1=0
(α4)2=1
α=5 or 3.

Take focus (3,0)
Equation of directrix lx+my=1 satisfies (0,3) i.e. mirror image of focus through tangent y=x
0+3m=1
m=13

Slope of line joining (0,3) and (4,4) is 14 which is parallel to axis.
Equation of axis y0=14(x3)
x4y3=0

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