Perpendicular Distance of a Point from a Plane

Q.

Find the equation of the ellipse satisfying the given condition $e=\frac{3}{4}$, foci on Y-axis, centre at origin and passes through (6, 4).

Or
Find the equation of the hyperbola with vertices at $(\pm 5,0)$ and foci $(\pm 7,0)$

Q. The length of the perpendicular from the origin to the plane passing through three non-collinear points $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ is

1. $\frac{\left|\right[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\left]\right|}{|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|}$
2. $\frac{2\left|\right[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\left]\right|}{|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|}$
3. $\left|\right[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\left]\right|$
4. $\frac{\left|\right[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\left]\right|}{2|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|}$

Q.

Find the set of real values of x for which ${\log }_{(x+3)}$ ($x^2$ - x) < 1________.

1. (-1, 0) U (-3, -2)

2. (-3, 3)

3. (-1, 3) U (-3, -2)

4. (-1, 0) U (1, 3) U (-3, -2)

Q.

Find the cartesian form of equation of the straight line $z(1-i)+\overline{z}(1+i)$ = 4

1. x - y = 2

2. x + y = 1

3. x + y = 2

4. x - y = 1

Q. Consider a plane $x+y-z=1$ and point $A(1,2,-3).$ A line $L$ has the equation $x=1+3r,y=2-r$ and $z=3+4r.$

The distance between the points on the line which are at a distance of $4\sqrt{3}$ from the plane is

1. $4\sqrt{26}$
2. $20$
3. $10\sqrt{13}$
4. none of these

Q. If the points $(1,1,\lambda )$ and $(-3,0,1)$ are equidistant from the plane, $3x+4y-12z+13=0,$ then $\lambda$ satisfies the equation:

1. $3x^2-10x+21=0$
2. $3x^2+10x-13=0$
3. $3x^2-10x+7=0$
4. $3x^2+10x+7=0$

Q.

What will be the perpendicular distance of a point P (7, 5 , -2) from the plane -2x +7y - 4z + 5 = 0 ?

1. $\frac{23}{\sqrt{69}}$

2. $\frac{34}{\sqrt{69}}$

3. $\frac{57}{\sqrt{69}}$

4. $\frac{62}{\sqrt{69}}$

Q. Two systems of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and a', b', c' from the origin, then

1. $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{a^{\prime 2}}+\frac{1}{b^{\prime 2}}+\frac{1}{c^{\prime 2}}=0$
2. $\frac{1}{a^2}+\frac{1}{b^2}-\frac{1}{c^2}+\frac{1}{a^{\prime 2}}+\frac{1}{b^{\prime 2}}-\frac{1}{c^{\prime 2}}=0$
3. $\frac{1}{a^2}-\frac{1}{b^2}-\frac{1}{c^2}+\frac{1}{a^{\prime 2}}-\frac{1}{b^{\prime 2}}-\frac{1}{c^{\prime 2}}=0$
4. $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{1}{a^{\prime 2}}-\frac{1}{b^{\prime 2}}-\frac{1}{c^{\prime 2}}=0$