A sphere of radius \(r=3\) is centred at \(C(6,-3,2)\).
A line passes through \(A(3,-1,6)\) and \(B(5,-1,-5)\).
Show that this line is a tangent to the sphere. (4 marks)
--- 12 WORK AREA LINES (style=lined) ---
Aussie Maths & Science Teachers: Save your time with SmarterEd
A sphere of radius \(r=3\) is centred at \(C(6,-3,2)\).
A line passes through \(A(3,-1,6)\) and \(B(5,-1,-5)\).
Show that this line is a tangent to the sphere. (4 marks)
--- 12 WORK AREA LINES (style=lined) ---
\(\text{See Worked Solutions}\)
\(\text{The sphere centred at} \ \ C(6,-3,2) \ \ \text {with radius} \ \ r=3\)
\((x-6)^2+(y+3)^2+(z-2)^2=3^2=9\)
\(\text{The line}\ A B\ \text{has direction}\)
\(\overrightarrow{O B}-\overrightarrow{O A}=\left[\begin{array}{c}5-3 \\ -1-(-1) \\ -5-6\end{array}\right]=\left[\begin{array}{c}2 \\ 0 \\ -11\end{array}\right]\)
\(\text{Line \(A B\) has parametric equation:}\)
\(\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\overrightarrow{O A}+\lambda \overrightarrow{A B}=\left[\begin{array}{c}3 \\ -1 \\ 6\end{array}\right]+\lambda\left[\begin{array}{c}2 \\ 0 \\ -11\end{array}\right]=\left[\begin{array}{c}3+2 \lambda \\ -1 \\ 6-11 \lambda\end{array}\right]\)
\(\text{Substitute into the equation for the sphere:}\)
| \((3+2 \lambda-6)^2+(-1+3)^2+(6-11 \lambda-2)^2\) | \(=9\) |
| \((2 \lambda-3)^2+4+(4-11 \lambda)^2\) | \(=9\) |
| \(9-12 \lambda+4 \lambda^2+4+16-88 \lambda+121 \lambda^2\) | \(=9\) |
| \(125 \lambda^2-100 \lambda+20\) | \(=0\) |
| \(5\left(25 \lambda^2-20 \lambda+4\right)\) | \(=0\) |
| \(5(5 \lambda-2)^2\) | \(=0\) |
\(\text{There is only one}\ \lambda\ \text{that solves this equation.}\)
\(\text{i.e. one common point on the line and the sphere.}\)
\(\therefore\ \text{The line is a tangent to the sphere.}\)