For all α, β∈R and αβ>0, the line αx+βy+αβ=0 is such that it

(A) possesses a slope independent of α and β

(B) passes through a fixed point

(D) possesses intercepts on the axes that differ by a quantity independent of α, β

TS EAMCET · Maths · Straight Lines

For all $α, β \in R$ and $α β > 0,$ the line $α x + β y + \sqrt{α β} = 0$ is such that it

A possesses a slope independent of $α$ and $β$
B passes through a fixed point
C forms a triangle of constant area with coordinate axes
D possesses intercepts on the axes that differ by a quantity independent of $α, β$

Verified Solution

Answer & Solution

Correct Answer

(C) forms a triangle of constant area with coordinate axes

Step-by-step Solution

Detailed explanation

αx+βy+αβ=0 divided by αβ αx+βy+αβαβ=0 ⇒ αβx+βαy=-1 The line intersect y-axis at 0,-αβ and x-axis at -βα,0 So area of the triangle formed by…

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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