First slide

Equation of line in Straight lines

Question

$Suppose A, B are two points on 2 x - y + 3 = 0 and P (1, 2) is such that P A = P B . Then the mid-point of A B is$

Easy

A

$(- \frac{1}{5}, \frac{13}{5})$
B

$(\frac{- 7}{5}, \frac{9}{5})$
C

$(\frac{7}{5}, \frac{- 9}{5})$
D

$(\frac{- 7}{5}, \frac{- 9}{5})$

Solution

In an Isosceles triangle median is the altitude
$\begin{aligned} Equation of A B = 2 x - y + 3 = 0 \\ \Rightarrow Δ P A D ≅ △ P B D \end{aligned}$
$\begin{aligned} \Rightarrow D is foot of perpendicular \Rightarrow from P to A B \\ \begin{aligned} \frac{α - 1}{2} = \frac{β - 2}{- 1} = \frac{- (2 \times 1 - 1 \times 2 + 3)}{4 + 1} \\ \frac{α - 1}{2} = \frac{β - 2}{- 1} = \frac{- 3}{5} \Rightarrow α = \frac{- 1}{5}, β = \frac{13}{5} \end{aligned} \end{aligned}$

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