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Fill in the Blanks: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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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 Fill in the Blanks: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand one focus at the point  Fill in the Blanks: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  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. Fill in the Blanks: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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

Fill in the Blanks: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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

Fill in the Blanks: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Fill in the Blanks: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

which is the standard equation of the ellipse.

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