Question

Sol. [Key Concept: A line through (2,3) cuts a chord of length 2 between the parallels $2x + y = 3$ and $2x + y = 5$.]

Slope of line is $m = tan(\alpha \pm \theta)$

$$tan\alpha = \frac{1}{2} \cdot tan\theta = 2 $$

$${m = \frac{\frac{1}{2} + 2}{1 - \frac{1}{2} \times 2} \rightarrow \infty }{m = \frac{\frac{1}{2} - 2}{1 + \frac{1}{2} \times 2} = \frac{- 3}{4} }$$eqn. of lines are

$x = 2$ and $y - 3 = \frac{- 3}{4}(x - 2)$

$x = 2$ and $4y + 3x = 18$

✘ (a) $y = 3$ is horizontal; the chord it makes between the parallels is not 2.

✔ (b) $x = 2$ (vertical) cuts a chord of length exactly 2.

✘ (c) $y = x + 1$ does not give a chord of length 2.

✔ (d) $4y + 3x = 18$ (slope $- \frac{34}{}$) cuts a chord of length 2.