Question

Find the image of the point (1, 2) in the line x − 2y − 7 = 0.

Answer 1

To find the image (x₂, y₂) of a point (x₁, y₁) in the line ax + by + c = 0, using the standard reflection formula:

`(x_2 - x_1)/a = (y_2 - y_1)/b = (-2(ax_1 + by_1 + c))/(a^2 + b^2)`

Substituting given values:

For the point (1, 2) and the line x − 2y − 7 = 0:

x₁ = 1, y₁ = 2, a = 1, b = −2, c = −7

Let’s first calculate the term in the numerator:

ax₁ + by₁ + c = 1(1) + (−2)(2) − 7

= 1− 4 − 7

= −10

Then, calculate the denominator:

a² + b² = 1² + (−2)²

= 1 + 4

= 5

Substitute these into the right side of the formula:

`(-2 (-10))/5 = 20/5`

= 4

Now, solve the two separate equations for the image coordinates:

⇒ For x₂:

`(x_2 - 1)/1 = 4`

x₂ − 1 = 4

∴ x₂ = 5

⇒ For y₂:

`(y_2 - 2)/2 = 4`

y₂ − 2 = −8

∴ y₂ = −6

Hence, the image of the point (5, −6).