If the equation of the plane bisecting the li | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The mid-point $R$ of the line segment joining $P(3,2,4)$ and $Q(-1,0,-2)$ is given by: $R = \left( \frac{3-1}{2}, \frac{2+0}{2}, \frac{4-2}{2} \ri

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) The mid-point $R$ of the line segment joining $P(3,2,4)$ and $Q(-1,0,-2)$ is given by: $R = \left( \frac{3-1}{2}, \frac{2+0}{2}, \frac{4-2}{2} \right) = (1, 1, 1)$. The direction ratios of the line segment $PQ$ are $(3 - (-1), 2 - 0, 4 - (-2)) = (4, 2, 6)$. Since the plane is perpendicular to $PQ$,the normal vector to the plane is $\vec{n} = \langle 4, 2, 6 \rangle$. Thus,the equation of the plane is $4x + 2y + 6z + d = 0$. Since the plane passes through the mid-point $R(1, 1, 1)$,we have: $4(1) + 2(1) + 6(1) + d = 0 \implies 4 + 2 + 6 + d = 0 \implies d = -12$. Comparing $4x + 2y + 6z - 12 = 0$ with $ax + by + cz + d = 0$,we get $a=4, b=2, c=6, d=-12$. Therefore,$ac + bd = (4)(6) + (2)(-12) = 24 - 24 = 0$.

If the equation of the plane bisecting the line segment joining the points $P(3,2,4)$ and $Q(-1,0,-2)$ and perpendicular to $PQ$ is $ax+by+cz+d=0$,then find the value of $ac+bd$.

Similar Questions

$O$ is the origin and $A$ is $(a, b, c)$. Find the direction cosines of the line $OA$ and the equation of the plane passing through $A$ at a right angle to $OA$.

Find the vector equation of the plane passing through the point $2\hat{i} + \hat{j} - 4\hat{k}$ and parallel to the plane $\vec{r} \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) - 7 = 0$.

The Cartesian equation of the plane $r = (1 + \lambda - \mu )i + (2 - \lambda )j + (3 - 2\lambda + 2\mu )k$ is

Find the Cartesian equation of the plane passing through the point with position vector $2\hat{i} + 3\hat{j} - 4\hat{k}$ and perpendicular to the vector $2\hat{i} - \hat{j} + 2\hat{k}$.

The direction ratios of the normal to the plane passing through $(1, 0, 0)$ and $(0, 1, 0)$ which makes an angle $\frac{\pi}{4}$ with the plane $x + y = 3$ are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module