Question

Let \(x=2t,\ y=\frac {t^2}{3}\) be a conic. Let S be the focus and B be the point on the axis of the conic such that SA⊥BA, where A is any point on the conic. If k is the ordinate of the centroid of the ΔSAB, then \(\lim\limits_{t \to 1} k\) is equal to

• A. \(\frac {17}{18}\)
• B. \(\frac {19}{18}\)
• C. \(\frac {11}{18}\)
• D. \(\frac {13}{18}\)

Answer 1

The Correct Option is D

\(x=2t,\ y=\frac 23\)
For \(t=1\)
\(A=(2,\frac 13)\)
Conic is \(x^2 = 12y \)\(⇒ S = (0, 3)\)
Let \(B = (0, β)\)
Given \(SA⊥BA\)
\((\frac {\frac 13}{2−3})(\frac {β−\frac 13}{−2})=−1\)

\((β−\frac 13)\frac 13=−2\)
\(β=\frac 13(−\frac {17}{3})\)
Ordinate of centroid,
\(K = \frac {β+\frac 13+3}{3}\)

\(=\frac {−\frac {17}{9}+\frac {10}{3}}{3}\)

\(= \frac {13}{18}\)

So, the correct option is (D): \(\frac {13}{18}\)