Find a relation between $x$ and $y$ such that the point $(x, y)$ is equidistant from the points $(7, 1)$ and $(3, 5)$.

Step 1: Define the points

Let the point $(x, y)$ be denoted as $P$. Let the two given points be $A(7, 1)$ and $B(3, 5)$. We are given that point $P$ is equidistant from $A$ and $B$, which means the distance $PA$ is equal to the distance $PB$.

Step 2: Set up the distance equality

Since point $P$ is equidistant from $A$ and $B$, their distances must be equal. To avoid dealing with square roots, we can square both sides of the equation. This leads to $PA^2 = PB^2$.

Step 3: Apply the distance formula

We use the distance formula to express $PA^2$ and $PB^2$. The distance formula states that the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Squaring this removes the square root, giving us $(x_2 - x_1)^2 + (y_2 - y_1)^2$.

Step 4: Equate the squared distances

Now we equate the expressions for $PA^2$ and $PB^2$ as established in the previous steps. This equation forms the basis for finding the relation between $x$ and $y$.

Step 5: Expand the terms

We expand each squared term using the algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$. This allows us to remove the parentheses and prepare for simplification.

Step 6: Simplify the equation

Combine the constant terms on each side. We have $49 + 1 = 50$ on the left side and $9 + 25 = 34$ on the right side. We also observe that $x^2$ and $y^2$ terms are present on both sides, and they will cancel out.

Step 7: Cancel common terms

We cancel the $x^2$ and $y^2$ terms from both sides of the equation, as they are identical. This simplifies the equation significantly, leaving only linear terms in $x$ and $y$ and constant terms.

Step 8: Rearrange and solve for the relation

Now, we gather all the $x$ terms and $y$ terms on one side and the constant terms on the other side. This rearrangement allows us to isolate the variables and simplify the equation into the desired relation between $x$ and $y$. Finally, divide by $8$ to get the simplest form.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

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