Find a point on the $y$ -axis which is equidistant from the points $A(6, 5)$ and $B(-4, 3)$.

Step 1: Define the point on the y-axis

A point on the $y$ -axis always has its $x$ -coordinate equal to 0. So, we can represent the required point as $P(0, y)$.

Step 2: Set up the distance condition

The problem states that the point $P$ is equidistant from points $A(6, 5)$ and $B(-4, 3)$. This means the distance from $P$ to $A$ is equal to the distance from $P$ to $B$.

Step 3: Apply the distance formula

To simplify calculations, we square both sides of the distance equality, $PA = PB$, to remove the square roots. We then apply the distance formula, which states that the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Step 4: Substitute the coordinates into the formula

Substitute the coordinates of $P(0, y)$, $A(6, 5)$, and $B(-4, 3)$ into the squared distance formula. For $PA^2$, we use $x_1=0, y_1=y$ and $x_2=6, y_2=5$. For $PB^2$, we use $x_3=0, y_3=y$ and $x_4=-4, y_4=3$.

Step 5: Simplify and expand the equation

First, calculate the squared terms for the $x$ -coordinates: $(-6)^2 = 36$ and $(4)^2 = 16$. Then, expand the squared binomials using the formula $(a-b)^2 = a^2 - 2ab + b^2$: $(y-5)^2 = y^2 - 10y + 25$ and $(y-3)^2 = y^2 - 6y + 9$.

Step 6: Combine constant terms and variables

Combine the constant terms on both sides of the equation. On the left side, $36 + 25 = 61$. On the right side, $16 + 9 = 25$.

Step 7: Solve for y

Subtract $y^2$ from both sides of the equation. Then, isolate the terms containing $y$ on one side and the constant terms on the other side. Perform the arithmetic to find the value of $y$.

Step 8: State the coordinates of the point

Since we found $y = 9$ and the point is on the $y$ -axis, its coordinates are $(0, 9)$.

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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