Equation of a Plane : Intercept Form

Q.

The equation of the plane perpendicular to $z$ - axis and passing through $\left(2,-3,5\right)$ is

1. $x-2=0$

2. $y+3=0$

3. $z-5=0$

4. $2x-3y+5z+4=0$

Q. The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ meets the coordinate axes at A, B, C respectively. D and E are the mid-points of AB and AC respectively. Coordinates of the mid-point of DE are

1. $(a,\frac{b}{4},\frac{c}{4})$
2. $(\frac{a}{4},b,\frac{c}{4})$
3. $(\frac{a}{4},\frac{b}{4},c)$
4. $(\frac{a}{2},\frac{b}{4},\frac{c}{4})$.

Q.

The distance of a point $\mathrm{P}\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)$ from the x-axis is

1. $\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}$

2. $\sqrt{{\mathrm{b}}^2+{\mathrm{c}}^2}$

3. $\mathrm{a}$

4. $\sqrt{{\mathrm{a}}^2+{\mathrm{c}}^2}$

Q.

If equation of a plane is given $4x+2y+12z=7$ then $x,y\&z$ intercepts will be

1. $\frac{7}{2},\frac{7}{4}\&\frac{7}{12}$

2. $\frac{7}{4},\frac{7}{2}\&\frac{7}{12}$

3. $\frac{4}{7},\frac{2}{7}\&\frac{12}{7}$

4. $\frac{2}{7},\frac{4}{7}\&\frac{12}{7}$

Q.

A variable plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ at a unit distance from origin cuts the coordinate axes at A, B and C. Centroid (x, y, z) satisfies the equation $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=K$ is

1. 9

2. 3

3. $\frac{1}{9}$

4. $\frac{1}{3}$

Q.

A variable plane moves so that the sum of reciprocals of its intercepts on the three coordinate axes is constant $\lambda .$ It passes through a fixed point, which has coordinates

1. $(\lambda ,\lambda ,\lambda )$

2. $(\frac{1}{\lambda },\frac{1}{\lambda }.\frac{1}{\lambda })$
   

3. $(-\lambda ,-\lambda ,-\lambda )$

4. $(-\frac{1}{\lambda },-\frac{1}{\lambda },-\frac{1}{\lambda })$

Q.

If the product of perpendiculars from the foci upon the polar of $P$ be constant and equal to $c^2$, prove that the locus of $P$ is the ellipse $b^2{x}^2(c^2+a^2{e}^2)+c^2{a}^4{y}^2=a^4{b}^4$.

Q. Find the equation of the plane with intercepts 2, 3 and 4 on the x, y and z-axes respectively

1. 6x+4y+3z = 12
2. 4x+3y-6z = 2
3. x+y+z = 1
4. 7x+2y+9z = 10

Q. A variable plane at a distance of $1$ unit from the origin cuts the coordinates axes at $A,B$ and $C.$ If the centroid $D(x,y,z)$ of triangle $ABC,$ satisfies the relation $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=k,$ then the value of $k$ is

1. $3$
2. $1$
3. $\frac{1}{3}$
4. $9$

Q.

A variable plane moves so that the sum of reciprocals of its intercepts on the three coordinate axes is constant $\lambda .$ It passes through a fixed point, which has coordinates

1. $(\lambda ,\lambda ,\lambda )$

2. $(\frac{1}{\lambda },\frac{1}{\lambda }.\frac{1}{\lambda })$
   

3. $(-\lambda ,-\lambda ,-\lambda )$

4. $(-\frac{1}{\lambda },-\frac{1}{\lambda },-\frac{1}{\lambda })$

Q. If a plane meets the co-ordinate axes in A, B, C such that the centroid of the triangle ABC is the point $(1,r,r^2),$ then equation of the plane is

1. $x+ry+r^{2}z=3r^2$
2. $r^{2}x+ry+z=3r^2$
3. $x+ry+r^{2}z=3$
4. $r^{2}x+ry+z=3$

Q. Points (1, 2, 3), (9, 5, 8), (-2, 5, 6) and (-3, -8, 8) are in the same plane.

1. True
2. False