The area (in square units) of the triangle fo | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The given pair of straight lines is $x^2-3xy+2y^2=0$.Factoring the equation: $x^2-2xy-xy+2y^2=0$ $\Rightarrow x(x-2y)-y(x-2y)=0$ $\Rightarrow (x-y

Vedclass · Accepted Answer

$\frac{1}{12}$

Vedclass · Answer

(C) The given pair of straight lines is $x^2-3xy+2y^2=0$.Factoring the equation: $x^2-2xy-xy+2y^2=0$ $\Rightarrow x(x-2y)-y(x-2y)=0$ $\Rightarrow (x-y)(x-2y)=0$.This represents two lines: $L_1: x-y=0$ and $L_2: x-2y=0$.The third line is $L_3: x+y+1=0$.To find the vertices of the triangle,we find the intersection points of these lines:$1$. Intersection of $L_1$ and $L_2$: Solving $x=y$ and $x=2y$ gives $(0,0)$.$2$. Intersection of $L_1$ and $L_3$: Solving $x=y$ and $x+y+1=0$ gives $2x+1=0 \Rightarrow x=-\frac{1}{2}, y=-\frac{1}{2}$. Point is $(-\frac{1}{2}, -\frac{1}{2})$.$3$. Intersection of $L_2$ and $L_3$: Solving $x=2y$ and $x+y+1=0$ gives $2y+y+1=0$ $\Rightarrow 3y=-1$ $\Rightarrow y=-\frac{1}{3}, x=-\frac{2}{3}$. Point is $(-\frac{2}{3}, -\frac{1}{3})$.The area of the triangle with vertices $(0,0)$,$(-\frac{1}{2}, -\frac{1}{2})$,and $(-\frac{2}{3}, -\frac{1}{3})$ is given by:Area $= \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$Area $= \frac{1}{2} |0(-\frac{1}{2} - (-\frac{1}{3})) + (-\frac{1}{2})(-\frac{1}{3} - 0) + (-\frac{2}{3})(0 - (-\frac{1}{2}))|$Area $= \frac{1}{2} |0 + \frac{1}{6} - \frac{1}{3}| = \frac{1}{2} |-\frac{1}{6}| = \frac{1}{12}$ square units.

The area (in square units) of the triangle formed by $x+y+1=0$ and the pair of straight lines $x^2-3xy+2y^2=0$ is

Similar Questions

Write reactions to justify the amphoteric nature of aluminium.

If $l, m, n$ are in arithmetic progression,then the straight line $lx + my + n = 0$ will always pass through the point:

The value of $a$ such that $f(x) = \sqrt{3} \sin x - \cos x - 2ax + b$ decreases for all real values of $x$ is given by:

Honey bees also produce beeswax,which finds many uses in industry,such as:

In the $C.G.S.$ system,the magnitude of the force is $100 \ dynes$. In another system where the fundamental physical quantities are $kilogram$,$metre$,and $minute$,the magnitude of the force is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module