JEE Advanced (Fill in the Blanks): Conic Sections

Q.1. The point of intersection of the tangents at the ends of the latus rectum of the parabola y2 = 4x is....... (1994 -  2 Marks)

Ans. ( –1,  0)
Sol.  Given parabola is y2 = 4x; a = 1
Extremities of latus rectum are (1, 2) and (1, – 2) tangent to y2 = 4x at (1, 2) is y.2 = 2 (x + 1) i.e.  y = x + 1 ...(1)
Similarly tangent at (1, – 2) is, y = – x – 1 ...(2)
Intersection pt. of these tangents can be obtained by solving (1) and (2), which is (– 1, 0).

Q.2. An ellipse has eccentricity and one focus at the point    Its one directrix is the common tangent, nearer to the point P, to the circle x2 + y2 =1 and the hyperbola x2 – y2 =1. The equation of the ellipse, in the standard form, is............ (1996 - 2 Marks)

Ans.

Sol.  Rough graph  of x2 + y2 = 1 (circle) ...(1)
and x2 – y2 = 1 (hyperbola) ...(2)
is as shown below.

It is clear from graph that there are two common tangents to the curves (1) and (2) namely x = 1 and x = – 1 out of which x = 1 is nearer to pt. P.
Hence directrix of required ellipse is x – 1 = 0 Also e = 1/2, focus (1/2, 1) then equation of ellipse is given by

which is the standard equation of the ellipse.

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## Chapter-wise Tests for JEE Main & Advanced

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