Question

The line segment joining A(3,0),B(5,2) are rotated about a point A in anticlockwise sense through an angle π/4 and B move to C. If point D be the reflection of C in Y - axis then D=

Answer 1

Given :-

• The line segment joining A(3,0),B(5,2) are rotated about a point A in anticlockwise sense through an angle π/4 and B move to C.

To Find :-

• Reflection of Coordinates C under y - axis . i.e. D

Solution :-

Given that,

• Coordinates of A (3, 0)

and

• Coordinates of B (5, 2)

We know,

Slope of a line joining two points A (a, b) and B (c, d) is represented by m and is given by

$\bf :\longmapsto\:m = \dfrac{d - b}{c - a}$

Let Slope of AB be m, then

$\rm :\longmapsto\:m = \dfrac{2 - 0}{5 - 3}$

$\rm :\longmapsto\:m = \dfrac{2}{2}$

$\bf\implies \:m = 1$

Let assume that line AB makes an angle p with positive direction of x axis.

$\rm :\implies\:tanp = 1$

$\rm :\implies\:tanp = tan45\degree$

$\bf\implies \:p = 45\degree$

Since,

• Line AB is rotated through an angle of 45° with x - axis in anti-clockwise direction.

It means,

• Line AC makes an angle of 90° with positive direction of x - axis.

Now,

We know that

Equation of line which passes through the point (a, b) and having slope m, then equation of line is given by

$\bf :\longmapsto\:y - b = m(x - a)$

So,

Equation of AC which passes through the point (3, 0) and makes an angle of 90° with positive direction of x axis is

$\rm :\longmapsto\:y - 0 = tan90\degree (x - 3)$

$\rm :\longmapsto\:y = \dfrac{1}{0}(x - 3)$

$\bf\implies \:x = 3$

Let

• Coordinates of point C be (x, y).

As x = 3

Therefore,

• Coordinates of C = (3, y)

As

• AB is rotated through an angle of 45° and took new pisition AC.

It means

• Length of AB = Length of AC

$\rm :\longmapsto\:AB = AC$

$\rm :\longmapsto\: {AB}^{2} = {AC}^{2}$

$\rm :\longmapsto\: {(5 - 3)}^{2} + {(2 - 0)}^{2} = {(3 - 3)}^{2} + {(y - 0)}^{2}$

$\rm :\longmapsto\:4 + 4 = {y}^{2}$

$\rm :\longmapsto\:8 = {y}^{2}$

$\bf\implies \:y = \sqrt{8} = \sqrt{2 \times 2 \times 2} = 2 \sqrt{2}$

Hence,

$\bf :\longmapsto\:Coordinates \: of \: C = (3,2 \sqrt{2})$

Now,

We know that

Reflection of Coordinates C under y - axis is obtained by changing the sign of x - coordinate.

So,

$\bf :\longmapsto\:Coordinates \: of \: D= ( - 3, 2 \sqrt{2})$

Answer 2

Step-by-step explanation:

-3,2,root2

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