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.