<p>The planes \(\pi_1\) and \(\pi_2\) have equations \(3x+c(y+z)-2=0\) and \(\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})\) respectively, where&nbsp;\(c\) is a constant, and \(s\) and \(t\) are parameters. The point \(A(1,3,-2)\) lies in both planes.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;\(c=-1\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Show that the vector equation of the line of intersection of \(\pi_1\) and \(\pi_2\), line \(l\), is given by \(\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\alpha(\mathbf{i}-\mathbf{j}+4\mathbf{k})\), where&nbsp;\(\alpha\) is a parameter.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the position vectors of the points on the line \(l\) which are a distance of \(3\sqrt{2}\) from the point \(B(2,-3,7)\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the equation of the plane \(\pi_3\) which is parallel to \(\pi_2\) and contains the point \(B\).</div><div class="list-item-paragraph">Hence show that the distance between the planes \(\pi_2\) and \(\pi_3\) is \(\frac{20}{3\sqrt{3}}\).<span class="inline-marks">[3]</span></div></li></ol>

The planes \(\pi_1\) and \(\pi_2\) have equations \(3x+c(y+z)-2=0\) and \(\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})\) respectively, where \(c\) is a constant, and \(s\) and \(t\) are parameters. The point \(A(1,3,-2)\) lies in both planes.

Show that the vector equation of the line of intersection of \(\pi_1\) and \(\pi_2\), line \(l\), is given by \(\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\alpha(\mathbf{i}-\mathbf{j}+4\mathbf{k})\), where \(\alpha\) is a parameter.

Find the position vectors of the points on the line \(l\) which are a distance of \(3\sqrt{2}\) from the point \(B(2,-3,7)\).

Find the equation of the plane \(\pi_3\) which is parallel to \(\pi_2\) and contains the point \(B\).

Hence show that the distance between the planes \(\pi_2\) and \(\pi_3\) is \(\frac{20}{3\sqrt{3}}\).

2025 ACJC P1 Q9

2025 ACJC P1 Q9