First slide

Pair of straight lines

Question

If the line passing through P(1, 2) making an angle $\frac{π}{4}$ with the x-axis in the positive direction meets the pair of lines
$x^{2} + 4 x y + y^{2} at A and B, then P A \cdot P B =$

Moderate

A

13/3
B

13/6
C

11/6
D

11/3

Solution

$x = x_{1} \pm r \cos θ, y = y_{1} \pm r \sin θ$
$x = 1 \pm \frac{r}{\sqrt{2}}, y = 2 \pm \frac{r}{\sqrt{2}}$
$\begin{array}{l} Put (1 + \frac{r}{\sqrt{2}}, 2 + \frac{r}{\sqrt{2}}) in x^{2} + 4 x y + y^{2} = 0 \\ 1 + \frac{r^{2}}{2} + \frac{2 r}{\sqrt{2}} + 4 (2 + \frac{2 r}{\sqrt{2}} + \frac{r}{\sqrt{2}} + r^{2}) + 4 + \frac{r^{2}}{2} + \frac{4 r}{\sqrt{2}} = 0 \\ 3 r^{2} + 9 \sqrt{2} r + 13 = 0 {is a quadratic equation in r(distance) for which product of the roots (r}_{1} r_{2} or PA.PB ) = \frac{13}{3} \end{array}$

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