Question

If the acute angle between the two regression lines is $\theta$, then

• A. ${\theta }_1={\theta }_2$
• B. ${\theta }_1=2{\theta }_2$
• C. $2{\theta }_1={\theta }_2$
• D. none of these

Answer 1

The correct option is A

${\theta }_1={\theta }_2$

Explanation for Correct Option

Step1:Comparing given equation with the general equation

General equation $=a{x}^2+2hxy+b{y}^2=0$

The angle between the two lines is $\tan \theta =\left|\frac{\left(2\sqrt{(h2–ab)}\right)}{(a+b)}\right|$

Given equation $3x^2–7xy+4y^2=0....\left(\mathrm{i}\right)$

Comparing equation (i) with the general equation

$a=3,h=\frac{-7}{2},b=4$

$$\begin{gathered} \tan {\theta }_1=\left|\frac{\left(2\sqrt{\left({\displaystyle \frac{49}{4}}–12\right)}\right)}{(3+4)}\right| \\ \tan {\theta }_1=\left|\frac{2\sqrt{{\displaystyle \frac{1}{4}}}}{7}\right| \\ \tan {\theta }_1=\frac{1}{7}....\left(ii\right) \end{gathered}$$

Step 2: Repeat the same process with $2^{nd}$ equation

Second given equation $=6x^2–5xy+y^2=0.......\left(iii\right)$

Comparing the equation $\left(iii\right)$ with the general equation

$a=6,h=\frac{-5}{2},b=1$

$$\begin{gathered} \tan {\theta }_1=\left|\frac{\left(2\sqrt{\left({\displaystyle \frac{25}{4}–6}\right)}\right)}{(6+1)}\right| \\ \tan {\theta }_1=\frac{1}{7}....\left(iv\right) \end{gathered}$$

From equations $\left(ii\right)$ and $\left(iv\right)$ we get,

${\theta }_1={\theta }_2$

Hence option (1) is Correct.