**How to Find Image of a Point About the Line :**

Here we are going to see how to find image of a point about the line.

**Question 1 :**

Find the image of the point (−2, 3) about the line x + 2y − 9 = 0.

**Solution :**

Let us draw a perpendicular line from the point (-2, 3).

Now we have to find the equation of the line perpendicular to the line x + 2y - 9 = 0.

Let us find the line which is perpendicular to the line x + 2y - 9 = 0.

2x - y + k = 0

The perpendicular line 2x - y + k = 0 is passing through the point (-2, 3)

2(-2) - 3 + k = 0

-4 - 3 + k = 0

k = 7

2x - y + 7 = 0

By solving the above two equations, we get the point B.

x + 2y - 9 = 0 -----(1)

2x - y + 7 = 0 -----(2)

Multiply the first equation by 2, we get

2x + 4y - 18 = 0

(1) - (2)

(2x + 4y - 18) - (2x - y + 7) = 0

2x + 4y - 18 - 2x + y - 7 = 0

5y - 25 = 0

y = 5

By applying the value of y in the first equation, we get

x + 2(5) - 9 = 0

x + 10 - 9 = 0

x + 1 = 0

x = -1

B(-1, 5)

Mid point of AA' = (x_{1} + x_{2})/2, (y_{1} + y_{2})/2

(-2 + a) /2 , (3 + b)/2 = (-1, 5)

(-2 + a)/2 = -1 -2 + a = -2 a = -2 + 2 a = 0 |
(3 + b)/2 = 5 3 + b = 10 b = 10 - 3 b = 7 |

Hence the required point is (0, 7)

**Question 2 :**

A line is drawn perpendicular to 5x = y + 7. Find the equation of the line if the area of the triangle formed by this line with co-ordinate axes is 10 sq. units.

**Solution :**

5x - y - 7 = 0

The line which is perpendicular to the above line is 1x + 5y + k = 0

By applying A(x, 0), we get the value of x in terms of k.

1x + 5(0) + k = 0

x + k = 0

x = -k

By applying B(0, y), we get the value of y in terms of k.

1(0) + 5y + k = 0

5y + k = 0

y = -k/5

here, x = base of triangle, y = height of triangle

Area of triangle = 10 square units

(1/2) ⋅ Base ⋅ Height = 10

(1/2) ⋅ (-k) ⋅ (-k/5) = 10

k^{2} = 100 ==> k = ± 10

x + 5y = ± 10

After having gone through the stuff given above, we hope that the students would have understood "How to Find Image of a Point About the Line".

Apart from the stuff given above, if you want to know more about "How to Find Image of a Point About the Line".

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