From the point (–1, –1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 0 and cx + dy = 7, a, b, c, dZ, then the value of ad + bc is

Question

From the point (–1, –1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 0 and cx + dy = 7, a, b, c, d $\in$ Z, then the value of ad + bc is ________. [2026]

Smart Achievers · Accepted Answer

(7)

We have, $\tan 45 ° = | \frac{m + 1}{1 - m} |$

$\Rightarrow \pm 1 = \frac{m + 1}{1 - m}$

$\Rightarrow$ m = 0 & one line is perpendicular to x-axis.

Rays are : $L_{1} : x = - 1, L_{2} : y = - 1$

Intersection points of $L_{1}$ and $L_{2}$ with x + 2y = 1 are (–1, 1) and (3, –1) respectively.

Now, taking mirror image of x + 1 = 0 in x + 2y = 1.

$A' = \frac{x + 1}{1} = \frac{y + 1}{2} = \frac{- 2 (1 - 2 - 1)}{1^{2} + 2^{2}} = \frac{8}{5}$

$\Rightarrow x = \frac{3}{5}, y = \frac{11}{5} and A' \equiv (\frac{3}{5}, \frac{11}{5})$

Now, taking mirror image of y + 1 = 0 in x + 2y = 1.

$A'' = \frac{x + 1}{1} = \frac{y + 1}{2} = \frac{- 2 (1 - 2 - 1)}{1^{2} + 2^{2}} = \frac{8}{5}$

$\Rightarrow x = \frac{3}{5}, y = \frac{11}{5}$

$A'' \equiv (\frac{3}{5}, \frac{11}{5})$

Equation of ray passes through (3, –1) is

$(y + 1) = \frac{(\frac{11}{5} + 1)}{(\frac{3}{5} - 3)} (x - 3)$

$\Rightarrow 4 x + 3 y = 9$

Comparing 4x + 3y = 9 with ax + by = 9, we get

a = 4, b = 3

Similarly, equation of ray passes through (–1, 1) is

–3x + 4y = 7

Comparing with cx + dy = 7, we get

c = –3, d = 4

Thus ad + bc = $4 \times 4 + 3 \times (- 3) = 16 - 9 = 7$

Solution

Q.
From the point (–1, –1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 0 and cx + dy = 7, a, b, c, d $\in$ Z, then the value of ad + bc is ________. [2026]

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Solution

Q. From the point (–1, –1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 0 and cx + dy = 7, a, b, c, d∈Z, then the value of ad + bc is ________. [2026]

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Q.
From the point (–1, –1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 0 and cx + dy = 7, a, b, c, d $\in$ Z, then the value of ad + bc is ________. [2026]