P is a fixed point (a, a, a) on a line throug | vedclass

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

Question

(D) The line passes through the origin $(0, 0, 0)$ and is equally inclined to the axes,so its direction ratios are $(1, 1, 1)$.The point $P$ is $(a, a

Vedclass · Accepted Answer

$\frac{1}{a}$

Vedclass · Answer

(D) The line passes through the origin $(0, 0, 0)$ and is equally inclined to the axes,so its direction ratios are $(1, 1, 1)$.The point $P$ is $(a, a, a)$. The vector $\vec{OP}$ is the normal to the plane,so the normal vector is $\vec{n} = a\hat{i} + a\hat{j} + a\hat{k}$,which is parallel to $\hat{i} + \hat{j} + \hat{k}$.The equation of the plane passing through $(a, a, a)$ with normal vector $(1, 1, 1)$ is:$1(x - a) + 1(y - a) + 1(z - a) = 0$$x + y + z = 3a$To find the intercepts,we write the equation in the intercept form $\frac{x}{X} + \frac{y}{Y} + \frac{z}{Z} = 1$:$\frac{x}{3a} + \frac{y}{3a} + \frac{z}{3a} = 1$The intercepts on the axes are $X = 3a$,$Y = 3a$,and $Z = 3a$.The sum of the reciprocals of the intercepts is:$\frac{1}{3a} + \frac{1}{3a} + \frac{1}{3a} = \frac{3}{3a} = \frac{1}{a}$.

$P$ is a fixed point $(a, a, a)$ on a line through the origin equally inclined to the axes. Then,any plane through $P$ perpendicular to $OP$ makes intercepts on the axes,the sum of whose reciprocals is equal to:

Similar Questions

If the equation of the plane passing through the point $(3,2,5)$ and perpendicular to the planes $2x-3y+5z=7$ and $5x+2y-3z=11$ is $x+by+cz+d=0$,then $2b+3c+d=$

If the plane $\frac{x}{2} + \frac{y}{3} + \frac{z}{3} = 1$ intersects the coordinate axes at $A, B, C$,then the area of $\Delta ABC$ is:

In space,the equation $by + cz + d = 0$ represents a plane perpendicular to the plane:

The equation of the plane in Cartesian form,which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and its normal vector drawn from the origin being $2 \hat{i}-3 \hat{j}+4 \hat{k}$,is

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three non-coplanar vectors,then the vector equation $\overrightarrow{r}=(1-p-q) \overrightarrow{a}+p \overrightarrow{b}+q \overrightarrow{c}$ represents a :

Vedclass Test Series

Exam Paper Generator

Online Exam Module