If the distance of two points P and Q on the parabola y2=4 a x from the focus of a parabola are 4 and 9 respectively then the distance of the point of intersection of tangents at P and Q from the focus is

Question

If the distance of two points P and Q on the parabola y2=4 a x from the focus of a parabola are 4 and 9 respectively then the distance of the point of intersection of tangents at P and Q from the focus is (A) 8 (B) 6 (C) 3.5 (D) 13

Tardigrade Admin · Accepted Answer

Let T is the point of the intersection of the tangents at P, Q. We have S P=a+a t12, S Q=a+a t22 Also T ≡(a t1 t2, a(t1+t2)) Now S T2=(a-a t1 t2)2+(a(t1+t2))2 =a2(1+t1 2 t2 2+t1 2+t2 2) =a2(1+t12)(1+t22)=(a+a t12)(a+a t2 2) =S P × S Q=4 × 9=36 ∴ S T=6

Q. If the distance of two points $P$ and $Q$ on the parabola $y^{2} = 4 a x$ from the focus of a parabola are $4$ and $9$ respectively then the distance of the point of intersection of tangents at $P$ and $Q$ from the focus is

$8$

$6$

$3.5$

$13$

Q. If the distance of two points P and Q on the parabola y2=4ax from the focus of a parabola are 4 and 9 respectively then the distance of the point of intersection of tangents at P and Q from the focus is

8

6

3.5

13

Let T is the point of the intersection of the tangents at P,Q. We have SP=a+at12​,SQ=a+at22​ Also T≡(at1​t2​,a(t1​+t2​)) Now ST2=(a−at1​t2​)2+(a(t1​+t2​))2 =a2(1+t1​2t2​2+t1​2+t2​2) =a2(1+t12​)(1+t22​)=(a+at12​)(a+at2​2) =SP×SQ=4×9=36 ∴ST=6

Q. If the distance of two points $P$ and $Q$ on the parabola $y^{2} = 4 a x$ from the focus of a parabola are $4$ and $9$ respectively then the distance of the point of intersection of tangents at $P$ and $Q$ from the focus is

$8$

$6$

$3.5$

$13$