Question

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.

Hint:

Using slopes is often faster and less prone to expanding binomial mistakes:
\[ m_{AB} = \frac{1 - 0}{1 - a} = \frac{1}{1 - a} \] \[ m_{BC} = \frac{b - 1}{0 - 1} = 1 - b \] Since \(AB \perp BC\): \[ \frac{1 - b}{1 - a} = -1 \implies 1 - b = a - 1 \implies a + b = 2 \] Both methods yield the identical, clean result!

Answer 1

Correct Answer: 2

Step 1: Understanding the Question:
This question is from Coordinate Geometry.
We are given three points \(A(a, 0)\), \(B(1, 1)\), and \(C(0, b)\) which form a right-angled triangle with the right angle at vertex \(B\).
We need to establish an algebraic relationship between the variables \(a\) and \(b\).

Step 2: Key Formula or Approach:
We can solve this problem in two ways:
1. Pythagoras Theorem: Since the triangle is right-angled at \(B\), we must have: \[ AC^2 = AB^2 + BC^2 \] 2. Slope Method (recommended for speed): Since \(AB \perp BC\), the product of their slopes must equal \(-1\): \[ m_{AB} \times m_{BC} = -1 \] where the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Detailed Explanation:
Let us use the Pythagoras Theorem method to find the relation:
1. Calculate the squared distances using the distance formula \(d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2\): - Distance \(AB^2\): \[ AB^2 = (1 - a)^2 + (1 - 0)^2 \] \[ AB^2 = (1 - a)^2 + 1 = 1 - 2a + a^2 + 1 = a^2 - 2a + 2 \] - Distance \(BC^2\): \[ BC^2 = (0 - 1)^2 + (b - 1)^2 \] \[ BC^2 = 1 + (b - 1)^2 = 1 + b^2 - 2b + 1 = b^2 - 2b + 2 \] - Distance \(AC^2\): \[ AC^2 = (0 - a)^2 + (b - 0)^2 \] \[ AC^2 = a^2 + b^2 \] 2. Since the triangle is right-angled at \(B\), apply Pythagoras' theorem: \[ AC^2 = AB^2 + BC^2 \] 3. Substitute the calculated squared distances: \[ a^2 + b^2 = (a^2 - 2a + 2) + (b^2 - 2b + 2) \] 4. Simplify the equation by cancelling common terms on both sides: Subtract \(a^2 + b^2\) from both sides: \[ 0 = -2a - 2b + 4 \] 5. Rearrange the terms to isolate the variables: \[ 2a + 2b = 4 \] Divide the entire equation by 2: \[ a + b = 2 \]

Step 4: Final Answer:
The relation between \(a\) and \(b\) is \(a + b = 2\).