Q.(a) Show that the pair of straight lines $4x^2+12xy+9y^2-6x-9y+2=0$ represents two parallel straight lines and also find the separate equations of the straight lines.

(a) Put $u=2x+3y$: $u^2-3u+2=0\Rightarrow(u-1)(u-2)=0$, giving parallel lines $2x+3y-1=0,\ 2x+3y-2=0$. (b) Means from the intersection $(\bar X,\bar Y)=(13,17)$; $r=\sqrt{b_{yx}b_{xy}}=0.6$.

Part (a) — Pair of parallel straight lines

Given: $4x^2+12xy+9y^2-6x-9y+2=0$.

Step 1 — recognise the perfect square.

$4x^2+12xy+9y^2=(2x+3y)^2.$

Also $-6x-9y=-3(2x+3y)$. Let $u=2x+3y$:

$u^2-3u+2=0.$

Step 2 — factorise.

$(u-1)(u-2)=0\;\Rightarrow\;u=1\text{ or }u=2.$

Step 3 — write the separate lines.

$2x+3y-1=0\qquad\text{and}\qquad 2x+3y-2=0.$

Both have the same coefficients of $x$ and $y$ ($2$ and $3$), so they are parallel. (Condition check: $a=4,b=9,h=6\Rightarrow h^2=36=ab$, confirming parallel lines.)

Part (b) — Means and correlation from regression lines

Given regression equations: $4X-5Y+33=0$ and $20X-9Y-107=0$.

Step 1 — the lines intersect at $(\bar X,\bar Y)$. Solve:

$4X-5Y=-33\quad(\times5):\ 20X-25Y=-165.$

Subtract from $20X-9Y=107$:

$(20X-9Y)-(20X-25Y)=107-(-165)\Rightarrow16Y=272\Rightarrow Y=17.$

Then $4X-5(17)=-33\Rightarrow4X=52\Rightarrow X=13.$ So $\bar X=13,\ \bar Y=17.$

Step 2 — identify the regression coefficients. …