TS EAMCET · Maths · Straight Lines

The equation of the straight line in the normal form which is parallel to the lines $x + 2 y + 3 = 0$ and $x + 2 y + 8 = 0$ and dividing the distance between these two lines in the ratio $1 : 2$ internally is

A $x \cos α + y \sin α = \frac{10}{\sqrt{45}}, α = \tan^{- 1} \sqrt{2}$
B $x \cos α + y \sin α = \frac{14}{\sqrt{45}}, α = π + \tan^{- 1} 2$
C $x \cos α + y \sin α = \frac{14}{\sqrt{45}}, α = \tan^{- 1} 2$
D $x \cos α + y \sin α = \frac{10}{\sqrt{45}}, α = π + \tan^{- 1} \sqrt{2}$

Verified Solution

Answer & Solution

Correct Answer

(B) $x \cos α + y \sin α = \frac{14}{\sqrt{45}}, α = π + \tan^{- 1} 2$

Step-by-step Solution

Detailed explanation

Let the required equation of the line be x+2y+k=0 ...1 The distance between two lines x+2y+3=0 and x+2y+8=0 divide 1:2 internally. ⇒|k-3|8-3:|k-8|8-3=1:2 ⇒|k-3|5:|k-8|5=1:2 ⇒|k-3||k-8|=12 ⇒2|k-3|=|k-8| ⇒22k-32=k-82…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The equation of the straight line in the normal form which is parallel to the lines x+2y+3=0 and x+2y+8=0 and dividing the distance between these two lines in the ratio 1:2 internally is

Step-by-step Solution

The equation of the straight line in the normal form which is parallel to the lines $x + 2 y + 3 = 0$ and $x + 2 y + 8 = 0$ and dividing the distance between these two lines in the ratio $1 : 2$ internally is