The equation 4x^2 + 12xy + 9y^2 + 2gx + 2fy + | vedclass

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

Question

(C) The general equation of second degree $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of parallel lines if $h^2 = ab$ and $af^2 = bg^2$

Vedclass · Accepted Answer

$g = 2, f = 3, c$ is any number

Vedclass · Answer

(C) The general equation of second degree $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of parallel lines if $h^2 = ab$ and $af^2 = bg^2$.Comparing the given equation with the general form,we have $a = 4, h = 6, b = 9$.Check condition $h^2 = ab$: $6^2 = 4 	imes 9 \Rightarrow 36 = 36$,which is satisfied.Check condition $af^2 = bg^2$: $4f^2 = 9g^2$ $\Rightarrow (2f)^2 = (3g)^2$ $\Rightarrow 2f = 3g$ $\Rightarrow f = \frac{3}{2}g$.If we take $g = 2$,then $f = 3$.The condition for the equation to represent a pair of lines is $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0$.Substituting the values: $4 	imes 9 	imes c + 2(3)(2)(6) - 4(3^2) - 9(2^2) - c(6^2) = 0$.$36c + 72 - 36 - 36 - 36c = 0$.$0 = 0$.This identity holds true for any value of $c$. Thus,$g = 2, f = 3$ and $c$ is any number.

The equation $4x^2 + 12xy + 9y^2 + 2gx + 2fy + c = 0$ represents two real parallel straight lines,if

Similar Questions

If $\varphi$ is the angle between the lines $a x^{2}+2 h x y+b y^{2}=0$,then the angle between the lines $x^{2}+2 x y \sec \theta+y^{2}=0$ is

If the lines represented by $(p - q) x^2 + 2 (p + q) xy + (q - p) y^2 = 0$ are perpendicular to each other,then:

If $\theta$ is the angle between the lines represented by $x^2 - 3xy + \lambda y^2 + 3x - 5y + 2 = 0,$ where $\lambda$ is a real number,then $\csc^2 \theta$ equals:

The angle between the pair of straight lines $3y^2 - 8xy - 3x^2 - 29x + 3y - 18 = 0$ is (in $^{\circ}$)

If the slope of one of the lines represented by $5x^2 + \frac{40}{3}xy + ky^2 = 0$ is $3$,then the angle between the pair of lines is

Vedclass Test Series

Exam Paper Generator

Online Exam Module