Question

The line $r=a+tb$ touches the sphere $r^2-2r.c+h=0,c^2>h$ at the point with position vector $a$ if

• A. $(a-b).c=0$
• B. $(a-c).b=0$
• C. $(b-c).a=0$
• D. $a.b+b.c+c.a=0$

Answer 1

Answer: (B)

$(a-c).b=0$

Given $r=a+tb$ touches the sphere $r^2-2rc+h=0$

$r^2-2rc+h=0$

Add $c^2$ on both sides

$(r-c{)}^2=c^2-h$

centre of sphere is $c$

Let $r=a+tb$ touches sphere at $a$

d.r's of line joining $a$ and $c$ is $(a-c)$

d.r's of $r=a+tb$ is $b$ as both are perpendicular

Dot product of d.r's $=0$

$(a-c)b=0$