Coordinate Geometry
RD Sharma
CBSE
Grade 10
Question:
If the distance of $P(x, y)$ from $A(5, 1)$ and $B(-1, 5)$ are equal, prove that $3x = 2y$.
Step-by-Step Solution
Key Concept: $(x-5)^2 + (y-1)^2 = (x+1)^2 + (y-5)^2 \Rightarrow x^2 - 10x + 25 + y^2 - 2y + 1 = x^2 + 2x + 1 + y^2 - 10y + 25 \Rightarrow -10x - 2y = 2x - 10y \Rightarrow 8y = 12x \Rightarrow 3x = 2y$.
$(x - 5)^2 + (y - 1)^2 = (x + 1)^2 + (y - 5)^2$. [0.5 Mark]
$-10x + 26 - 2y = 2x + 26 - 10y \Rightarrow 8y = 12x$. [1.0 Mark]
$3x = 2y$. Proved! [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Setting $PA^2 = PB^2$: 0.5 Mark
Simplifying to $8y = 12x$: 1.0 Mark
Concluding $3x = 2y$: 0.5 Mark
Correct Answer:
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