TS EAMCET · Maths · Pair of Lines

The ratio in which the line $x + y - 1 = 0$ divides the line segment joining the origin and the point of intersection of the lines represented by $2 x^{2} - 13 x y - 7 y^{2} + x + 23 y - 6 = 0$ is

A $15 : 11$
B $- 11 : 15$
C $7 : 3$
D $7 : 19$

Verified Solution

Answer & Solution

Correct Answer

(A) $15 : 11$

Step-by-step Solution

Detailed explanation

2x2-13xy-7y2+x+23y-6=0 Point of intersection ∂f∂x=4x-13y+1=0 ∂f∂y=-14y-13x+23=0 ⇒4x13x+14y-23=0 ⇒13x(4x-13y+1)=0 ⇒225y-105=0 y=715 4x=9115-1 x=1915 P(1915,715) ∂=-0+0-11915+715-1 ∂=1511

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 ratio in which the line x+y-1=0 divides the line segment joining the origin and the point of intersection of the lines represented by 2x2-13xy-7y2+x+23y-6=0 is

Step-by-step Solution

The ratio in which the line $x + y - 1 = 0$ divides the line segment joining the origin and the point of intersection of the lines represented by $2 x^{2} - 13 x y - 7 y^{2} + x + 23 y - 6 = 0$ is