Question

If the slope of one of the two lines given by \(\frac{x^2}{a}+\frac{2xy}{h}+\frac{y^2}{b} = 0\) is twice that of the other, then \(h^2:ab =\) ...

• A. \(1:2\)
• B. \(2:1\)
• C. \(8:9\)
• D. \(9:8\)

Hint:

Put y = mx to get a quadratic in m, then use sum and product of roots with slopes m and 2m.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Both lines pass through the origin, so each has the form \(y=mx\). Putting \(y=mx\) in the pair equation gives a quadratic in \(m\) whose two roots are the slopes.

Step 2: Form the quadratic in slopes:
\[ \frac{x^2}{a}+\frac{2mx^2}{h}+\frac{m^2x^2}{b}=0 \Rightarrow \frac{m^2}{b}+\frac{2m}{h}+\frac{1}{a}=0 \]

Step 3: Use the given condition:
Let the slopes be \(m\) and \(2m\).
Product of roots: \(2m^2=\frac{1/a}{1/b}=\frac{b}{a}\), so \(m^2=\frac{b}{2a}\).
Sum of roots: \(3m=-\frac{2/h}{1/b}=-\frac{2b}{h}\).

Step 4: Eliminate m:
Square the sum relation: \(9m^2=\frac{4b^2}{h^2}\). Replace \(m^2\) by \(\frac{b}{2a}\):
\[ \frac{9b}{2a}=\frac{4b^2}{h^2} \Rightarrow h^2=\frac{8ab}{9} \]
So \(h^2:ab=8:9\).

Step 5: Why the other options are wrong:
The ratios \(1:2\) and \(2:1\) only repeat the slope ratio and ignore the coefficients. \(9:8\) is the inverted ratio, which comes from writing \(\frac{ab}{h^2}\) instead of \(\frac{h^2}{ab}\).

Final Answer:
The required ratio is \(8:9\), which is option (C). \[ \boxed{8:9} \]