The planes ${{p}_{1}}$ and ${{p}_{2}}$ have equations $x-4y+8z=4$ and $mx+ny+2z=1$ respectively, whe

Question

The planes ${{p}_{1}}$ and ${{p}_{2}}$ have equations $x-4y+8z=4$ and $mx+ny+2z=1$ respectively, where $m$ and $n$ are constants. If ${{p}_{1}}$ and ${{p}_{2}}$ meet at a line that has equation $\mathbf{r}=2\mathbf{i}-0.5\mathbf{j}+\lambda \left( -4\mathbf{i}+\mathbf{j}+\mathbf{k} \right)$, where $\lambda \in \mathbb{R}$, find the values of $m$ and $n$. [3] It is given instead that $m=1$ and $n=2$. Find the acute angle between ${{p}_{1}}$ and ${{p}_{2}}$. [2] The point $\left( 1,b,5 \right)$ is equidistant from ${{p}_{1}}$ and ${{p}_{2}}$. Calculate the possible value(s) of $b$. [6]

Accepted Answer

(i) $4m - n = 2$; (ii) $70.5°$; (iii) $b = -\frac{67}{2}$ \text{ or } $b = \frac{7}{10}$

2017 YJC P2 Q5

2017 YJC P2 Q5

Video Solution:

Solution:

Solution locked