TS EAMCET · Maths · Straight Lines

If the family of straight lines $a x + b y + c = 0$ , where $2 a + 3 b = 4 c$ is concurrent at the point $P (l, m)$ , then the foot of the perpendicular drawn from $P$ to the line $x + y + 1 = 0$ is

A $(\frac{- 3}{8}, \frac{- 5}{8})$
B $(\frac{- 2}{5}, \frac{- 3}{5})$
C $(3, - 4)$
D $(- 5, 4)$

Verified Solution

Answer & Solution

Correct Answer

(A) $(\frac{- 3}{8}, \frac{- 5}{8})$

Step-by-step Solution

Detailed explanation

Line ax+by+c=0 is compared with -a2-3b4+c=0 x=-12, y=-34 ⇒P-12,-34 Now, Foot of ⊥ ⇒x+121=y+341=--12-34+12 ⇒x+12=y+34=-18 ⇒x=-38, y=-58

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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If the family of straight lines ax+by+c=0, where 2a+3b=4c is concurrent at the point Pl, m, then the foot of the perpendicular drawn from P to the line x+y+1=0 is

Step-by-step Solution

If the family of straight lines $a x + b y + c = 0$ , where $2 a + 3 b = 4 c$ is concurrent at the point $P (l, m)$ , then the foot of the perpendicular drawn from $P$ to the line $x + y + 1 = 0$ is