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Question

L1 & L2 are the two lines which passes through the point (1,−1) and tangent to the curve y=x2−3x+2. Then which of the following point(s) lie on the hyperbola having L1 & L2 as its asymptotes and passing through the origin?

A
(2,2)
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B
(23,2)
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C
(83,0)
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D
(2,2)
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Solution

The correct options are
A (2,2)
B (23,2)
C (83,0)
The equation of line from (1,1) having slope m is y=mxm1
This line is tangent to the curve y=x23x+2.
mxm1=x23x+2x2(m+3)x+m+3=0=0(m+3)24(m+3)=0
m=3 or m=1
The equation of tangents (L1 & L2) are 3x+y2=0 & xy2=0
The equation of hyperbola and combined equation of its asymptotes differs only by constant.
The equation of hyperbola (3x+y2)(xy2)+λ=0
It passes through (0,0)λ=4
(3x+y2)(xy2)4=0

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