Point on X-axis Equidistant Calculation
The objective is to locate a specific point situated on the X-axis. This point must satisfy the condition of being equidistant from two given points, namely A(2, -3) and B(-2, 1).
Characteristics of a Point on the X-axis
A fundamental property of any point lying on the X-axis is that its y-coordinate is always zero. Consequently, we can represent the unknown point as P(x, 0).
Equidistant Condition Explained
The core requirement is that the distance from P to A is equal to the distance from P to B. This is expressed as $PA = PB$. Squaring both sides simplifies the calculation, giving us $PA^2 = PB^2$.
Applying the Distance Formula
The distance formula calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ as $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
Calculating the squared distance $PA^2$:
$PA^2 = (x - 2)^2 + (0 - (-3))^2$ $PA^2 = (x - 2)^2 + (3)^2$ $PA^2 = (x - 2)^2 + 9$Calculating the squared distance $PB^2$:
$PB^2 = (x - (-2))^2 + (0 - 1)^2$ $PB^2 = (x + 2)^2 + (-1)^2$ $PB^2 = (x + 2)^2 + 1$Solving the Equation for x
Equating the squared distances $PA^2$ and $PB^2$:
$(x - 2)^2 + 9 = (x + 2)^2 + 1$Expanding the squared terms:
$(x^2 - 4x + 4) + 9 = (x^2 + 4x + 4) + 1$Simplifying both sides:
$x^2 - 4x + 13 = x^2 + 4x + 5$Subtract $x^2$ from both sides to eliminate it:
$-4x + 13 = 4x + 5$Rearranging the terms to isolate x:
$13 - 5 = 4x + 4x$ $8 = 8x$Solving for x:
$x = \frac{8}{8}$ $x = 1$Final Point Coordinates
We determined that $x = 1$. Since the point P lies on the X-axis, its coordinates are P(x, 0). Substituting the value of x, we get P(1, 0).
Thus, the point P(1, 0) is equidistant from points A(2, -3) and B(-2, 1).
Similar Questions
- \(A(1, 2)\), \(B(a, -26)\), \(C(13, -14)\) and \(D(6, 14)\) are the vertices of a parallelogram, taken in order, find the value of \(a\).
- What would be the area of the triangle with \(A(8, 8)\), \(B(22, 8)\) and \(C(16, -10)\) as vertices?
- What would be the point which divides the line segment connecting the points \(P(10, 18), Q(5, 8)\) in the ratio \(2 : 3\) internally
- What is the midpoint of the line segment joining \((-2, -1)\) and \((-5, -3)\)?
- If point C is equidistant from A(5, -6) and B(7, 8), coordinate of C will be
- The area of the triangle whose vertices are given by (2, 4), (– 3, – 1) and (5, 3) is:
Important Questions from Coordinate Geometry
Find the co-ordinates of the centroid of a triangle whose vertices are A(1, 4), B(7, 8) and C(10, 12).
- If x² + y² - 16x + 38y + 425 = 0, then the value of x² + y² is:
- If x² + y² - 12x + 18y + 117 = 0, then the value of x² + y² is:
- A geological survey team has established three research stations forming a triangular monitoring zone. The stations are located at coordinates P(2,5), Q(8,1), and R(4,9) on a coordinate grid where each unit represents 1 kilometer. What is the area of the triangular monitoring zone?
- \(A(1, 2)\), \(B(a, -26)\), \(C(13, -14)\) and \(D(6, 14)\) are the vertices of a parallelogram, taken in order, find the value of \(a\).
