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Area of Triangle Using Coordinates

If the coordi...

Question

If the coordinates of two points A and B are (3,4) and (5,-2) respectively. Then, the coordinates of any point P, if PA = PB and area of $Δ$ PAB = 10, are

(7, 5), (1, 0)

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(7, 2), (1, 0)

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(7, 2), (- 1, 0)

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(2, 7), (1, 0)

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Solution

The correct option is B
(7, 2), (1, 0)

$L e t t h e c o o r d i n a t e s o f P b e (x, y) . T h e n, P A = P B \Rightarrow P A^{2} = P B^{2} \Rightarrow (x - 3)^{2} + (y - 4)^{2} = (x - 5)^{2} + (y + 2)^{2} \Rightarrow x - 3 y - 1 = 0 \dots \dots \dots (i) N o w, A r e a o f Δ P A B = 10 \Rightarrow \frac{1}{2} ∣ ∣ ∣ ∣ \begin{matrix} x & y & 1 3 & 4 & 1 5 & - 2 & 1 \end{matrix} ∣ ∣ ∣ ∣ = \pm 10 \Rightarrow 6 x + 2 y - 26 = \pm 20 \Rightarrow 6 x + 2 y - 46 = 0 o r 6 x + 2 y - 6 = 0 \Rightarrow 3 x + y - 23 = 0 o r 3 x + y - 23 = 0 S o l v i n g x - 3 y - 1 = 0 a n d 3 x + y - 23 = 0 W e g e t x = 7, y = 2 S o l v i n g x - 3 y - 1 = 0 a n d 3 x + y = 30 W e g e t x = 1, y = 0 T h u s, t h e c o r d i n a t e s o f P a r e (7, 2) o r (1, 0) . H e n c e, (b) i s t h e c o r r e c t a n s w e r .$

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If the coordinates of two points A and B are (3,4) and (5,-2) respectively. Then, the coordinates of any point P, if PA = PB and area of Δ PAB = 10, are

If the coordinates of two points A and B are (3,4) and (5,-2) respectively. Then, the coordinates of any point P, if PA = PB and area of $Δ$ PAB = 10, are