TS EAMCET · Maths · Pair of Lines

If a line $L$ is common to the pairs of lines $6 x^{2} - x y - 12 y^{2} = 0$ and $15 x^{2} + 14 x y - 8 y^{2} = 0$ , then the combined equation the other two lines is

A $10 x^{2} - 19 x y + 6 y^{2} = 0$
B $5 x^{2} - 4 x y + 7 y^{2} = 0$
C $x^{2} - 9 x y + y^{2} = 0$
D $3 x^{2} + 6 x y + 11 y^{2} = 0$

Verified Solution

Answer & Solution

Correct Answer

(A) $10 x^{2} - 19 x y + 6 y^{2} = 0$

Step-by-step Solution

Detailed explanation

Solving the given pair of straaight line we get, 6x2-xy-12y2=0 ⇒6x2-9xy+8xy-12y2=0 ⇒3x2x-3y+4y2x-3y=0 ⇒3x+4y2x-3y=0 ...............1 Now solving the other pair of straight line we get, 15x2+14xy-8y2=0 ⇒15x2+20xy-6xy-8y2=0 ⇒5x3x+4y-2y3x+4y=0…

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 a line L is common to the pairs of lines 6x2-xy-12y2=0 and 15x2+14xy-8y2=0, then the combined equation the other two lines is

Step-by-step Solution

If a line $L$ is common to the pairs of lines $6 x^{2} - x y - 12 y^{2} = 0$ and $15 x^{2} + 14 x y - 8 y^{2} = 0$ , then the combined equation the other two lines is