A B, A C are tangents to a parabola y2=4 a x, if l1, l2, l3 are the lengths of perpendiculars from A, B, C on any tangents to the parabola, then

Question

A B, A C are tangents to a parabola y2=4 a x, if l1, l2, l3 are the lengths of perpendiculars from A, B, C on any tangents to the parabola, then (A) l1, l2, l3 are in GP (B) l2, l1, l3 are in GP (C) l3, l1, l2 are in GP (D) l3, l2, l1 are in GP

Tardigrade Admin · Accepted Answer

Let the coordinates of B and C be (a t12, 2 a t1) and (a t22, 2 a t2) respectively. Then, the coordinates of A are (a t1 t2, a(t1+t2)). The equation of any tangent to y2=4 a x is t y=x+a t2. l1=(a t1 t2-a(t1+t2) t+a t2/√1+t2) l2=(a t12-2 a t1+a t2/√1+t2) and l3=(a t22-2 a t2+a t2/√1+t2) Clearly, l2 l3=l12 Therefore, l2, l1, l3 are in GP.

Q. $A B, A C$ are tangents to a parabola $y^{2} = 4 a x$ , if $l_{1}, l_{2}, l_{3}$ are the lengths of perpendiculars from $A, B, C$ on any tangents to the parabola, then

$l_{1}, l_{2}, l_{3}$ are in GP

$l_{2}, l_{1}, l_{3}$ are in GP

$l_{3}, l_{1}, l_{2}$ are in GP

$l_{3}, l_{2}, l_{1}$ are in GP

Q. AB,AC are tangents to a parabola y2=4ax, if l1​,l2​,l3​ are the lengths of perpendiculars from A,B,C on any tangents to the parabola, then

l1​,l2​,l3​ are in GP

l2​,l1​,l3​ are in GP

l3​,l1​,l2​ are in GP

l3​,l2​,l1​ are in GP

Q. $A B, A C$ are tangents to a parabola $y^{2} = 4 a x$ , if $l_{1}, l_{2}, l_{3}$ are the lengths of perpendiculars from $A, B, C$ on any tangents to the parabola, then

$l_{1}, l_{2}, l_{3}$ are in GP

$l_{2}, l_{1}, l_{3}$ are in GP

$l_{3}, l_{1}, l_{2}$ are in GP

$l_{3}, l_{2}, l_{1}$ are in GP