1.

Lists I, II and III contains conics, equation of tangents to the conics and points of contact, respectively. If the tangent to a suitable conic (List I) at (sqrt3(1)/(2)) is found to be sqrt3x+2y=4. then which of the following options is the only CORRECT combination?

Answer»

(II) (iii) (R)
(IV) (iv) (S)
(IV) (iii) (S)
(II) (iv) (R)

Solution :Tangent is `(SQRT3,(1)/(2))` is
`sqrt3x+2y=4`
Since slope of tangent at `(sqrt3,(1)/(2))` is `-ve`, possible curves are (I) and (II) only (draw the diagram and verify).
ALSO, given equation of tangent cannot match with
`my=x^(2)x+a`.
So, comparing eq. (I) with `y=mx+asqrt(x^(2)+1)`, we get
`a=2` and `m=(-sqrt3)/(2)`.
THEREFORE, equation of CURVE is `(x^(2))/(4)+y=1`.
The corresponding point of CONTACT is
`((-a^(2)m)/(sqrt(a^(2)m^(2)+1)),(1)/(sqrt(a^(2)m^(2)+1)))`


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