Q1.Prove that the orthocentre of the triangle formed by any three tangents to the parabola lie on the directrix.
Q2.If a normal to a parabola makes an angle θ with the axis ,show that it will cut the curve again at an angle of tan-1((tanθ)/2).
242
take a general pt. P (at12,2at1)
the normal eqn at pt P is y = -t1x +2at1 +at13
=> tanθ=-t1=slope of normal -----------(1)
the normal meets curve again at pt. Q (at22,2at2)
=> t2=-t1- 2/t1 ------------(2)
Let angle b/w normal and parabola=α=angle bw normal and tangent at Q
Now use, tanα=m1-m2/1+m1m2
where m1= -t1 and m2= 1/t2
use (1) and (2) t oget the answer [1][1]
1Q.1)let the equations to 3 tangents be: y=mx + a/m , y=nx + a/n, , y=px + a/p
find out point of intersection of all three pair of equations.
then write the equation to straight line through that point perpendicular to the other equation.
finally,u would see, the point common to the straight lines of perpendiculars i.e. the orhocenter of the triangle has coordinates :
x=-a, y=a(1/m + 1/n + 1/p + 1/mnp),and this pt. lies on the directrix