Question

If the normals to the parabola $y^2=4ax$ at three points $P,Q$ and $R$ meet at $A$ and $S$ is the focus, then $SP\cdot SQ\cdot SR$ is equal to

• A. $(SA{)}^2$
• B. $(SA{)}^3$
• C. $a(SA{)}^2$
• D. $a(SA{)}^3$

Answer 1

Answer: (C)

$a(SA{)}^2$

Let the points $A$ be $(h,k)$

Normally $y=mx-2am-{am}^3$ passes through $(h,k)$

$k=mh-2am-{am}^3$

$P=({a{m}_1}^2,-2a{m}_1),Q=({a{m}_2}^2,-2a{m}_2),R=({a{m}_3}^2,-2a{m}_3)$

$SP=\sqrt{{(a-a{m_1}^2)}^2+({2a{m}_1)}^2}$

$=a\sqrt{{(1+{m_1}^2)}^2}$

$=a(1+{m_1}^2)$

$SP\ast SQ\ast SR=a^3\sqrt{{(1+{m_1}^2)}^2+{(1+{m_2}^2)}^2+{(1+{m_3}^2)}^2}$

$=a^3\sqrt{(1+\sum {m1}^2+\sum {m1}^2\ast {m2}^2+{m1}^2\ast {m2}^2{\ast m3}^2}$

By simplification it reduces to $a\ast {(b-a)}^2+k^2$

So we get $SP\ast SQ\ast SR=a{SA}^2$