Find the value of k if the point P(k, 3) is equidistant from the points A(1, 5) and B(7, 1… — JAMB UTME

Find the value of $k$ if the point $P(k, 3)$ is equidistant from the points $A(1, 5)$ and $B(7, 1)$ .

Cevap: 4

The value of

k

is 4.

Using the distance formula, the squared distance

PA^2 = (k-1)^2 + (3-5)^2 = (k-1)^2 + 4

, and

PB^2 = (k-7)^2 + (3-1)^2 = (k-7)^2 + 4

. Equating

PA^2 = PB^2

gives

(k-1)^2 = (k-7)^2

. Expanding both sides yields

k^2 - 2k + 1 = k^2 - 14k + 49

. Subtracting

k^2

from both sides gives

12k = 48

, which leads to

k = 4

Write the expressions for the squared distances

PA^2

and

PB^2

using the distance formula.

PA^2 = (k - 1)^2 + 4

and

PB^2 = (k - 7)^2 + 4

The distance formula between two points

(x_1, y_1)

and

(x_2, y_2)

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

Equate

PA^2

and

PB^2

since point

P

is equidistant from points

A

and

B

(k - 1)^2 + 4 = (k - 7)^2 + 4 \implies (k - 1)^2 = (k - 7)^2

Subtracting 4 from both sides simplifies the equality of squared distances.

Expand both sides and isolate

k

to find its numerical value.

k^2 - 2k + 1 = k^2 - 14k + 49 \implies 12k = 48 \implies k = 4

Canceling

k^2

terms yields a simple linear equation.

Equidistant points and the distance formula in coordinate geometry

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