MHT CET · Maths · Straight Lines
If \(\frac{x^2}{\mathrm{a}}+\frac{2 x y}{\mathrm{~h}}+\frac{y^2}{\mathrm{~b}}=0\) represents a pair of straight lines and slope of one of the lines is twice that of the other, then \(a b: h^2\) is
[Note: The question has been modified to get the correct answer.]
- A \(1: 2\)
- B \(9: 8\)
- C \(2: 1\)
- D \(8: 9\)
Answer & Solution
Correct Answer
(B) \(9: 8\)
Step-by-step Solution
Detailed explanation
\(\mathrm{m}_1: \mathrm{m}_2=1: 2\)
If the slopes of the lines given by \(\mathrm{a} x^2+2 \mathrm{~h} x y+\mathrm{b} y^2=0\) are in the ratio m:n, then \({ }^{-}\) \((\mathrm{m}+\mathrm{n})^2 \mathrm{ab}=4 \mathrm{mnh}^2\)
\(\therefore (1+2)^2\left(\frac{1}{a}\right)\left(\frac{1}{b}\right)=4(1)(2)\left(\frac{1}{h}\right)^2 \quad \Rightarrow \frac{a b}{h^2}=\frac{9}{8}\)
[Note: In the question, \(\frac{x^2}{\mathrm{a}^2}\) is changed to \(\frac{x^2}{\mathrm{a}}\) to apply appropriate textual concepts.]
If the slopes of the lines given by \(\mathrm{a} x^2+2 \mathrm{~h} x y+\mathrm{b} y^2=0\) are in the ratio m:n, then \({ }^{-}\) \((\mathrm{m}+\mathrm{n})^2 \mathrm{ab}=4 \mathrm{mnh}^2\)
\(\therefore (1+2)^2\left(\frac{1}{a}\right)\left(\frac{1}{b}\right)=4(1)(2)\left(\frac{1}{h}\right)^2 \quad \Rightarrow \frac{a b}{h^2}=\frac{9}{8}\)
[Note: In the question, \(\frac{x^2}{\mathrm{a}^2}\) is changed to \(\frac{x^2}{\mathrm{a}}\) to apply appropriate textual concepts.]
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