If point $A(x, y)$ is equidistant from $B(3, 6)$ and $C(-3, 4)$, find a direct linear relation connecting $x$ and $y$.

Step 1: Understand the Equidistant Condition

The problem states that point $A(x, y)$ is equidistant from point $B(3, 6)$ and point $C(-3, 4)$. This means the distance from A to B is equal to the distance from A to C.

Step 2: Apply the Distance Formula

To find the distances AB and AC, we use the distance formula. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Step 3: Set up the Equation

Now, we substitute the coordinates of points A, B, and C into the distance formula. For AB, we use $A(x, y)$ and $B(3, 6)$. For AC, we use $A(x, y)$ and $C(-3, 4)$. Since $AB = AC$, we set their distance formulas equal to each other.

Step 4: Square Both Sides and Expand

To eliminate the square roots, we square both sides of the equation. Then, we expand the squared terms using the algebraic identity $(a-b)^2 = a^2 - 2ab + b^2$ and $(a+b)^2 = a^2 + 2ab + b^2$.

Step 5: Simplify and Solve for the Relation

We cancel out the $x^2$ and $y^2$ terms from both sides. Then, we gather all terms on one side of the equation and simplify to find the linear relation between $x$ and $y$. We can divide the entire equation by -4 to get a simpler 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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