Question

For \(a,b,h>0\), if the slope of one of the lines represented by \[ a^2x^2+2hxy+b^2y^2=0 \] is twice that of the other, then the value of \[ \frac{h}{ab} \] is:

• A. \[ \frac{3\sqrt{2}}{4} \]
• B. \[ \frac{2\sqrt{3}}{4} \]
• C. \[ -\frac{2\sqrt{3}}{4} \]
• D. \[ -\frac{3\sqrt{2}}{4} \]

Hint:

For a homogeneous equation \(Ax^2+2Hxy+By^2=0\), the slopes of the two lines are obtained from \(Bm^2+2Hm+A=0\). Then use sum and product of roots.

Answer 1

The Correct Option is A

Step 1: Express the pair of lines in terms of slopes.
The equation a²x² + 2hxy + b²y² = 0 represents two lines through the origin. Substituting y = mx yields b²m² + 2hm + a² = 0, whose roots are the slopes m₁ and m₂.

Step 2: Apply the condition that one slope is double the other.
Let m₁ = 2m and m₂ = m. The product of roots gives m₁m₂ = a²/b², so 2m² = a²/b², yielding m = a/(b√2).

Step 3: Use the sum of roots to relate h and the slopes.
m₁ + m₂ = -2h/b². Substituting 3m = 2h/b² (taking magnitude since h>0) and the expression for m gives 3a/(b√2) = 2h/b². Multiplying by b² yields 3ab/√2 = 2h, so h = 3ab/(2√2).

Step 4: Compute the required ratio h/(ab).
h/(ab) = 3/(2√2). Rationalizing the denominator gives 3√2/4.

Step 5: Final conclusion.
The ratio h/(ab) equals 3√2/4.