Question

Find the value of a when the distance between the points (3, a) and (4, 1) is `sqrt10`

Answer 1

The distance d between two points `(x_1,y_1)` and `(x_2, y_2)` is given by the formula

`d = sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2)`

The distance between two points (3, a) and (4, 1) is given as `sqrt10`. Substituting these values in the formula for distance between two points we have

`sqrt10 = sqrt((3 - 4)^2 `+ (a - 1)^2)`

`sqrt10 = sqrt((-1)^2 + (a - 1))`

Now, squaring the above equation on both sides of the equals sign

`10 = (-1)^2 + (a - 1)^2`

`10 = 1 + (a^2 + 1 - 2a)`

`8 = a^2 - 2a`

Thus we arrive at a quadratic equation. Let us solve this now,

`a^2 - 2a - 8 = 0`

`a^2 -4a + 2a - 8 = 0`

a(a - 4) + 2(a - 4) = 0

(a - 4)(a + 2) = 0

The roots of the above quadratic equation are thus 4 and −2.

Thus the value of ‘a’ could either be 4 or -2