The area of a triangle is 5. Two of its...

MathematicsArea of TriangleJEE Advanced 1978Easy

The area of a triangle is 5. Two of its vertices are $A (2, 1)$ and $B (3, - 2)$ . The third vertex $C$ lies on $y = x + 3$ . Find $C$ .

Answer:(\frac{7}{2}, \frac{13}{2}) \text{ or } (-\frac{3}{2}, \frac{3}{2})

Visualized Solution (English)

Visualizing the Given Data

Given vertices: $A (2, 1)$ and $B (3, - 2)$
Area of $△ A B C = 5$
Vertex $C$ lies on the line: $y = x + 3$

Defining Vertex .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C Parametrically

Let the $x$ -coordinate of $C$ be $λ$ .
Since $C$ lies on $y = x + 3$ , its $y$ -coordinate is $λ + 3$ .
Coordinates of $C = (λ, λ + 3)$

The Area Formula

Area $= \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣$
Substitute Area $= 5$ and coordinates of $A, B, C$ .

Substituting the Values

$\frac{1}{2} ∣2 (- 2 - (λ + 3)) + 3 ((λ + 3) - 1) + λ (1 - (- 2)) ∣ = 5$
Multiplying by $2$ : $∣2 (- 2 - λ - 3) + 3 (λ + 2) + λ (3) ∣ = 10$

Simplifying the First Term

First term: $2 (- 2 - λ - 3) = 2 (- λ - 5)$
Result: $- 2 λ - 10$

Simplifying the Second and Third Terms

Second term: $3 (λ + 3 - 1) = 3 (λ + 2) = 3 λ + 6$
Third term: $λ (1 - (- 2)) = λ (3) = 3 λ$

Combining the Terms

Summing terms: $(- 2 λ - 10) + (3 λ + 6) + 3 λ$
Grouping $λ$ : $(- 2 + 3 + 3) λ = 4 λ$
Grouping constants: $- 10 + 6 = - 4$
Equation: $∣4 λ - 4∣ = 10$

Handling the Absolute Value

Property: $∣ x ∣ = a \Rightarrow x = a$ or $x = - a$
Case 1: $4 λ - 4 = 10$
Case 2: $4 λ - 4 = - 10$

Solving Case 1

Case 1: $4 λ = 14 \Rightarrow λ = \frac{14}{4} = \frac{7}{2}$
$x = \frac{7}{2}$
$y = \frac{7}{2} + 3 = \frac{7 + 6}{2} = \frac{13}{2}$
Point $C_{1} = (\frac{7}{2}, \frac{13}{2})$

Solving Case 2

Case 2: $4 λ = - 10 + 4 = - 6 \Rightarrow λ = - \frac{6}{4} = - \frac{3}{2}$
$x = - \frac{3}{2}$
$y = - \frac{3}{2} + 3 = \frac{- 3 + 6}{2} = \frac{3}{2}$
Point $C_{2} = (- \frac{3}{2}, \frac{3}{2})$

Final Conclusion

The two possible coordinates for vertex $C$ are:
1. $(\frac{7}{2}, \frac{13}{2})$
2. $(- \frac{3}{2}, \frac{3}{2})$
Key Takeaway: Absolute value in area problems often yields two distinct geometric solutions.

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The area of a triangle is 5. Two of its vertices are A(2,1) and B(3,−2). The third vertex C lies on y=x+3. Find C.

Visualized Solution (English)

Visualizing the Given Data

Defining Vertex .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C Parametrically

The Area Formula

Substituting the Values

Simplifying the First Term

Simplifying the Second and Third Terms

Combining the Terms

Handling the Absolute Value

Solving Case 1

Solving Case 2

Final Conclusion

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