The values of \(y\) for which the distance between the points \(P(10,\,y)\) and \(Q(2,\,2)\) is 10 units are:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → -4 and 8
By the distance formula, \(PQ^{2}=(x_2-x_1)^{2}+(y_2-y_1)^{2}\). Here \(PQ=10\), so the square of the distance is 100. The unknown \(y\) appears squared, so the equation has two roots: one point P above Q and one below it, each 10 units away.
Putting in the coordinates: \((10-2)^{2}+(y-2)^{2}=100\)
Since \(8^{2}=64\): \(64+(y-2)^{2}=100\), so \((y-2)^{2}=36\)
Taking both square roots: \(y-2=6\) or \(y-2=-6\)
The two values: \(y=8\) or \(y=-4\)