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Question

If the tangents at P and Q in the parabola meet in T, then which of the following statements are correct.

1. TP and TQ subtends equal angle at the focus S

2. ST2 = SP.SQ


  1. Only 1

  2. Both 1 2

  3. None of these

  4. Only 2


Solution

The correct option is B

Both 1 2


Let's take a standard parabola y2=4ax and draw tangents

at P(at21,2at1) and Q(at22,2at2)... Which meet at T.

We know the point of intersection of tangent is T(at1t2,a(t1+t2))

We can also calculate coordinates of T by calculating tangents

at P & Q and their point of intersection is point T.

Tangent at P

y(2at1)=2a(x+at21)                  

yt1=x+at21                              ............(1)

Similarly tangent at Q

yt2=x+at22                              ............(2)

Solving equation (1) & (2)

We get, x=at1t2

             y=a(t1+t2)

Coordinates of T(at1t2,a(t1+t2)

Statement 1

To,check α=β, or check that T lies on the angle bisector

of the PSQ. i.e., perpendicular distance of T from the

line SP is equal to the perpendicular of T from SQ.

Equation of SP

y=2at2at21a(xa)

2t1x(t211)y2at1=0

P1=2at21t2(t211)a(t1+t2)2at1(t211)2+4t21

P1=a|t1t2|

Similarly equation of SQ

y=2at2at22a(xa)

2t2x(t221)2at2=0

Perpendicular distance of SQ from T

P2=2at22t1(t221)a(t1+t2)2at2(t221)2+4t22

So, we can say that α=β

Statement 1 is correct.

Statement 2:

ST2=(aat1t2)2+(a(t1+t2)0)2

          =a2(1t1.t2)2+a2(t1+t2)2

          =a2[1+t21t22+t21+t22]

SP=(at21a)2+(2at10)2=a2(t211)+4a2t21

     =at41+12t21+4t21=a(t21+1)2=a(t21+1)

SQ=(at22a)2+(2at10)2=a2(t22+1)+4a2t22

     =at42+12t22+4t22=a(t22+1)2=a(t22+1)

SP.SQ=a(t21+1)×a(t22+1)

                =a2[t21t22+t21+t22+1]

SP.SQ=a2[1+t21t22+t21+t22]=ST2

Statement 2 is correct

Both the statement are correct.

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