Q) Find a relation between x and y such that the point P(x, y) is equidistant from the points A (3, 5) and B (7, 1). Hence, write the coordinates of the points on x-axis and y-axis which are equidistant from points A and B.
PYQ: 29 – CBSE 2025 – Code 30 – Series 5 – Set 1
Ans:
Step 1: We know that the distance between two points (X1, Y1) and (X2, Y2) is given by:
S = √ [(X2 – X1)2 + (Y2 – Y1)2 ]
Now, distance between P (X, Y) and A (3, 5):
AP = √ [(3 – X)2 + (5 – Y)2 ] …………. (i)
Similarly, distance between B (7,1) and P (X, Y):
BP = √ [(X – 7)2 + (Y – 1)2 ] …………. (ii)
Step 2: Since P is equidistant from A & B,
∴ AP = BP
∴ √ [(3 – X)2 + (5 – Y)2 ] = √ [(X – 7)2 + (Y – 1)2 ]
∴ (3 – X)2 + (5 – Y)2 = (X – 7)2 + (Y – 1)2
∴ 9 – 6 X + X2 + 25 – 10 Y + Y2 = X2 – 14 X + 49 + Y2 – 2 Y + 1
∴ – 6 X – 10 Y + 34 = – 14 X – 2 Y + 50
∴ 14 X – 6 X – 10 Y + 2 Y = 50 – 34
∴ 8 X – 8 Y = 16
∴ X – Y = 2
Therefore, relation between X and Y is given by X – Y = 2.
Step 3: if point lies on X – axis, its ordinate value will be zero
∴ y = 0
By substituting this value of y in the relation equation, we get:
∵ X – Y = 2
∴ X – (0) = 2
∴ X = 2
Since this value is arrived, when y = 0,
∴ coordinate of a point on x – axis is (2, 0)
Step 4: Similarly, if point lies on Y – axis, its abscissa value will be zero
∴ x = 0
By substituting this value of x in the relation equation, we get:
∵ X – Y = 2
∴ (0) – Y = 2
∴ Y = – 2
Since this value is arrived, when x = 0,
∴ coordinate of a point on Y – axis is (0, – 2)
Therefore, coordinates of the points on x-axis and y-axis, equidistant from points A and B, are (2,0) and (0,2) respectively.
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