2023TrialShort answer

Q11 (d)

3 marks

Question

Find the equations of a sphere whose centre is at $(1,0,1)$ and touches the line

\vec r=\begin{pmatrix}3\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\2\end{pmatrix},\qquad \lambda\in\mathbb{R}.

Try the question before revealing the answer.

Solution

\begin{pmatrix}2+2\lambda\\1+\lambda\\1+2\lambda\end{pmatrix}\cdot\begin{pmatrix}3\\1\\2\end{pmatrix}=0

4+4\lambda+1+\lambda+2+4\lambda=0

9\lambda+7=0

\lambda=-\frac{7}{9}

Therefore, equation of sphere is

\left|\vec r-\begin{pmatrix}1\\0\\1\end{pmatrix}\right|=|\overrightarrow{CP}|

=\left|\begin{pmatrix}4/9\\2/9\\-5/9\end{pmatrix}\right|

\left|\vec r-\begin{pmatrix}1\\0\\1\end{pmatrix}\right|=\frac{\sqrt{5}}{3}

(x-1)^2+y^2+(z-1)^2=\frac{5}{9}

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Syllabus alignment

Vectors

Vectors and vector equations of linesUse coordinates and vector equations for curves, lines, line segments and spheres in two and three dimensions.
Further operations with three-dimensional vectorsUse vector magnitude, unit vectors and scalar products to solve problems and prove geometric results.

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