First slide

Equation of line in Straight lines

Question

Line passing through the point $P (2, 3)$ meets the lines represented by $x^{2} - 2 x y - y^{2} = 0$ at the points A and B such that $P A . P B = 17$ , the equation of the line is

Moderate

A

$x = 2$
B

$y = 3$
C

$3 x - 2 y = 0$
D

none of these

Solution

Let the equation of the line through $P (2, 3)$ making an angle $θ$ with the positive direction of x-axis be
$\frac{x - 2}{\cos θ} = \frac{y - 3}{\sin θ}$ .
Then the coordinates of any point on this line at a distance
r from p are $(2 + r \cos θ, 3 + r \sin θ)$ . If $P A = r_{1}$ and $P B = r_{2}$ , then $r_{1}, r_{2}$ are the roots of the equation.
$\begin{array}{l} (2 + r \cos θ)^{2} - 2 (2 + r \cos θ) (3 + r \sin θ) - (3 + r \sin θ)^{2} = 0 \\ \Rightarrow r^{2} (\cos 2 θ - \sin 2 θ) - 2 r (\cos θ + 5 \sin θ) - 17 = 0 \end{array}$
$\Rightarrow 17 = PA \cdot PB = r_{1} r_{2} = \frac{17}{\cos 2 θ - \sin 2 θ}$
$\Rightarrow \cos 2 θ - \sin 2 θ = 1$ which is satisfied by $θ = 0$ and thus
the equation of the line is $y = 3$ .

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