2 the value of k for which the point 0.2 is eqidistant from two points 3 k and (k 5 is)

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Find the value of k, if the point P 0, 2 equidistant from 3, k and k , 5.

Solution

P(0,2) Q(3,k) R(k,5)

PQ =

PR =

PQ =PR

=

Hence k =1 for P to be equidistant from Q and R

Mathematics

RD Sharma

Standard X

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)`

It is said that P(0,2) is equidistant from both A(3,k) and B(k,5).

So, using the distance formula for both these pairs of points we have

`AP =sqrt((3)^2 + (k - 2)^2)`

`BP = sqrt((k)^2 + (3)^2)`

Now since both these distances are given to be the same, let us equate both.

AP = Bp

`sqrt((3)^2 + (k -2)^2) = sqrt((k)^2 + (3)^2)`

Squaring on both sides we have,

`(3)^2 + (k - 2)^2 = (k)^2 + (3)^2`

`9 + k^2 + 4 - 4k = k^2 + 9`

4k = 4

k = 1

Hence the value of ‘k’ for which the point ‘P’ is equidistant from the other two given points is k = 1