First slide

Definition of a circle

Question

The equation of the chord of the circle $x^{2} + y^{2} - 3 x - 4 y - 4 = 0$ which passes through the origin such that the origin divides it in the ratio 4 : 1, is

Moderate

A

x = 0
B

$24 x + 7 y = 0$
C

$7 x + 24 y = 0$
D

$7 x - 24 y = 0$

Solution

Let y = mx be a chord.
Then the points of intersections are given by
$\begin{array}{l} x^{2} (1 + m^{2}) - x (3 + 4 m) - 4 = 0 \\ ∴ x_{1} + x_{2} = \frac{3 + 4 m}{1 + m^{2}} and x_{1} x_{2} = \frac{- 4}{1 + m^{2}} \end{array}$
Since (0, 0) divides chord in the ratio 1 : 4, we have
$\begin{array}{l} x_{2} = - 4 x_{1} \\ - 3 x_{1} = \frac{3 + 4 m}{1 + m^{2}} and 4 x_{1}^{2} = - \frac{- 4}{1 + m^{2}} \\ 9 + 9 m^{2} = 9 + 16 m^{2} + 24 m \\ i.e., m = 0, - \frac{24}{7} \end{array}$
Therefore, the lines are y= 0 and 7y + 24 x = 0.

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