Reflection of a Point about a Line

Q.

Which of the following point lies on the line 3x - 5y = 7?

1. (4, 3)

2. (3, 9)

3. (9, 4)

4. (8, 6)

Q. The point (4, 1)undergoes the following two successive transformation
(i) Reflection about the line y=x
(ii) Translation through a distance 2 units along the positive x-axis
Then the final coordinates of the point are

1. $(3,4)$
2. $(\frac{7}{2},\frac{7}{2})$
3. $(4,3)$
4. $(1,4)$

Q.

The point (4, 1) undergoes the following transformation successively.
(i) reflection about the line y = x
(ii) translation through a distance 2 units along the positive direction of x - axis
(iii) rotation through an angle $\frac{\pi }{4}$ about the origin in the anticlockwise direction.
(iv) reflection aout x = 0
The final position of the given point is

1. $(\frac{1}{\sqrt{2}},\frac{7}{2})$

2. $(\frac{1}{2},\frac{7}{\sqrt{2}})$

3. $\left(\frac{1}{\sqrt{2},}\frac{7}{\sqrt{2}}\right)$

4. $\left(\frac{1}{2}\frac{7}{2}\right)$

Q. The direction ratios of two perpendicular lines are $1,-3,5$ and $\lambda ,1+\lambda ,2+\lambda$ Then $\lambda$ is

1. $-\frac{7}{3}$
2. $-\frac{7}{2}$
3. $-\frac{1}{4}$
4. $-\frac{1}{2}$

Q. Let $0<\alpha <\frac{\pi }{2}$ be fixed angle. If
$P=(cos\theta ,sin\theta )$ and Q = $\left(cos\right(\alpha -\theta ),sin(\alpha -\theta \left)\right),$ then Q is obtained from P by -

1. cockwise rotation around origin through an angle $\alpha$
   
2. anticlockwise rotation around origin through an angle $\alpha$
   
3. reflection in the line throughorigin with slope tan $\alpha$
4. reflection in the line through origin with slope tan $\left(\alpha /2\right)$

Q. Which options is equal to $PX$ in the given figure?

1. $QY$
2. $2QY$
3. $\frac{1}{2}QY$
4. $XR$