First slide

Equation of line in Straight lines

Question

A line passing through the point P(4, 2), meets the x-axis and y-axis at A and B respectively. If O is the origin, then locus of the centre of the circum circle of $∆$ OAB is

Moderate

A

$x^{- 1} + y^{- 1} = 2$
B

$2 x^{- 1} + y^{- 1} = 1$
C

$x^{- 1} + 2 y^{- 1} = 1$
D

$2 x^{- 1} + 2 y^{- 1} = 1$

Solution

Let the coordinates of A and B be (a, 0) and (0, b) respectively.
Then, equation of line AB is $\frac{x}{a} + \frac{y}{b} = 1$
Since, it passes through the point P(4, 2)
$∴ \frac{4}{a} + \frac{2}{b} = 1$ -----(1)
Now, centre of the circumcircle of $∆ O A B = (\frac{a}{2}, \frac{b}{2})$
So, Eq. (1) can be written in the form $\frac{2}{a / 2} + \frac{1}{b / 2} = 1$
$∴$ locus of circumcentre is $\frac{2}{x} + \frac{1}{y} = 1$ or $2 x^{- 1} + y^{- 1} = 1$

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