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
Both 1 2
None of these
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
Similarly tangent at Q
Solving equation (1) & (2)
We get, x=at1t2
Coordinates of T(at1t2,a(t1+t2)
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
Similarly equation of SQ
Perpendicular distance of SQ from T
So, we can say that α=β
Statement 1 is correct.
Statement 2 is correct
Both the statement are correct.