Image of a Point with Respect to a Line

Q.

If the sum of the slopes of the lines given by $x^2-2cxy-7y^2=0$ is four times their product, then $c$ has the value:

1. $-2$

2. $-1$

3. $2$

4. $1$

Q. Let $a,b\in R.$ If the mirror image of the point $P(a,6,9)$ with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20,b,-a-9),$ then $|a+b|$ is equal to

1. $86$
2. $88$
3. $84$
4. $90$

Q.

Find the image of the point $P(2,-3,1)$ about the line
$\frac{x+1}{2}=\frac{y-3}{3}=\frac{z+2}{-1}$

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

2. $(\frac{-58}{7},\frac{36}{14},\frac{-40}{14})$

3. $(\frac{58}{7},\frac{36}{14},\frac{-40}{14})$

4. $(\frac{-58}{7},\frac{36}{14},\frac{40}{14})$

Q. If points $P(3,2,1),R(4,y,z),4Q(2,-1,3)$ lie on the same line, then distance of point R from origin is?

1. $\sqrt{31}$
2. $\sqrt{42}$
3. $2\sqrt{20}$
4. $\sqrt{21}$

Q.

The line passing through $A\left(-2,3\right)$ and $B\left(4,b\right)$ is perpendicular to the line $2x-4y=5$. Find the value of $b$.