Reflection of a Point about a Line

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

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

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

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

Q.

The angle made by the tangent to the circle $x^2+y^2-8x+6y+20$ = 0 at (3, -1) with the positive Direction of the x-axis is

1. 

2. 

3. 

4. 

Q.

Show that the points (a, 0), (0, b) and (3a, -2b) are collinear. Also find the equation of the line containing them.

Q. A ray of light along $x+\sqrt{3}y=\sqrt{3}$ gets reflected upon reaching X-axis, the equation of the reflected ray is

1. $\sqrt{3}y=x-\sqrt{3}$
2. $y=\sqrt{3}x-\sqrt{3}$
3. $y=x+\sqrt{3}$
4. $\sqrt{3}y=x-1$

Q.

The point $(4,1)$ undergoes the following three transformations successively

I. reflection about the line $y=x$.

II. translation through a distance of $2$ units along the positive direction of $x$-axis.

III. rotation through an angle of $\frac{\mathrm{\pi }}{4}$ about the origin in the anti-clockwise direction.

The final position of the point is:

Q. The point (4, 1) undergoes the following three transformations successively.
(i) Reflection about the line y = x
(ii) Transformation through a distance 2 unit along the positive direction of x-axis.
(iii) Rotation through an angle of π/4 about the origin in the anti-clockwise direction.
The final position of the point is given by the coordinates

1. 
2. 
3. 
4. 

Q. A ray of light passing through the point (1, 2) reflects on the x -axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

Q. Locus of image of the point $P(h,k)$ with respect to the line mirror which passes through the origin is

1. $x^2+y^2=hx+ky$
2. $x^2+y^2=h^2+k^2$
3. $x^2+y^2=h^2+hx$
4. $x^2+y^2=h^2+k^2+hx+ky$

Q.

Find the equation of the line passing through the point of intersection of the lines 4x + 7y - 3 = 0 and 2x - 3y + 1 = 0 that has equal intercepts on the axis.

Q.

Intersecting the y-axis at a distance of 2 units above the origin and making an angle of ${30}^{\circ }$ with positive direction of the x-axis.

Q. Find the image of the point (3, 8) with respect to the line x + 3 y = 7 assuming the line to be a plane mirror.

Q. Find the direction in which a straight line must be drawn through the point (1, 2) so that its point of intersection with the line x + y = 4 may be at a distance of $\sqrt{\frac{2}{3}}$ from this point.

Q.

The value of $co{s}^{-1}x+co{s}^{-1}(\frac{x}{2}+\frac{1}{2}\sqrt{3-3x^2})$ where $(\frac{1}{2}\le x\le 1)$ is equal to

1. $\frac{\pi }{3}$
2. $\frac{\pi }{6}$
3. $\pi$
4. $0$

Q. If the point (λ, λ + 1) lies inside the region bounded by the curve $x=\sqrt{25-y^2}$ and y-axis, then λ belongs to the interval
(a) (−1, 3)
(b) (−4, 3)
(c) (−∞, −4) ∪ (3, ∞)
(d) none of these

Q. Find the image of the point (2, 1) with respect to the line mirror x + y − 5 = 0.

Q. Statement 1: If $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ be the images of point $P(x,y)$ about lines $L_1\equiv ax+by+c$ and $L_2\equiv bx-ay+c^{\prime }=0$ respectively, then the line joining points $P_1$ and $P_2$ always passes through point of intersection of lines $L_1$ and $L_2$
Statement 2: Lines $L_1$ and $L_2$ are perpendicular

1. Statement $1$ is true, statement $2$ is also true; statement $2$ is the correct explanation of statement $1$
2. Statement $1$ is true, statement $2$ is also true; statement $2$ is not the correct explanation of statement $1$
3. Statement $1$ is true, statement $2$ is false
4. Statement $1$ is false, statement $2$ is true

Q. If the point $A$ is symmetric to the point $B(4,-1)$ with respect to the bisector of the first quadrant, then the length of $AB$ is

1. $5$ units
2. $5\sqrt{2}$ units
3. $3\sqrt{2}$ units
4. $3$ units

Q.

Match the given angles with the corresponding slopes.
$\begin{array}{cc}\text{Positive x direction}& \text{Slope of line}\\ {\text{1.) 0}}^{\text{o}}& \text{p.) 0}\\ {\text{2.) 90}}^{\text{o}}& \text{q.) -}\sqrt{3}\\ {\text{3.) 120}}^{\text{o}}& \text{r.) -}\frac{1}{\sqrt{3}}\\ {\text{4.) 150}}^{\text{o}}& \text{s.) Not defined}\end{array}$

1. 1 - p 2 - s 3 - r 4 - q

2. 1 - p 2 - q 3 - r 4 - s

3. 1 - s 2 - p 3 - q 4 - r

4. 1 - p 2 - s 3 - q 4 - r

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. Find the distance of the point (2, 5) from the line 3x + y + 4 = 0 measured parallel to a line having slope 3/4.

Q. A circle of radius 4 units touches the coordinate axes in the first quadrant. Find the equations of its images with respect to the line mirrors x = 0 and y = 0.

Q. The image of a point A(3, 8) in the line x+3y-7=0, is

1. (-1, -4)
2. (-3, -8)
3. (1, -4)
4. (3, 8)

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

Q. A ray of light along $x+\sqrt{3}y=\sqrt{3}$ gets reflected upon reaching X-axis, the equation of the reflected ray is

1. $y=x+\sqrt{3}$
2. $\sqrt{3}y=x-\sqrt{3}$
3. $y=\sqrt{3}x-\sqrt{3}$
4. $\sqrt{3}y=x-1$

Q. find the equation of circle whose end points of the diameter are A(-2, 3) and B(3, -5)

Q. The image of a point A(3, 8) in the line x+3y-7=0, is

1. (-1, -4)
2. (-3, -8)
3. (1, -4)
4. (3, 8)

Q. Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.

Q. The reflection of the point (4, -13) in the line 5x+y+6=0 is

1. (-1, -14)
2. (3, 4)
3. (1, 2)
4. (-4, 13)

Q. If the mirror image of point $(1,2)$ in the line $2y=x$ is $(a,b)$, then the value of $5(a+b)$ is:

Q. The equation of the hyperbola whose foci are (6, 4) and (−4, 4) and eccentricity 2, is

(a) $\frac{(x-1{)}^2}{25/4}-\frac{(y-4{)}^2}{75/4}=1$

(b) $\frac{(x+1{)}^2}{25/4}-\frac{(y+4{)}^2}{75/4}=1$

(c) $\frac{(x-1{)}^2}{75/4}-\frac{(y-4{)}^2}{25/4}=1$

(d) none of these