A pair of lines S=0 together with the lines g | vedclass

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

Question

(C) The given equation is $8 x^2-14 x y+3 y^2+10 x+10 y-25=0$.Factoring the quadratic part,we get $(4 x-y-5)(2 x-3 y+5)=0$.The lines are $L_1: 4 x-y-5

Vedclass · Accepted Answer

$8 x^2-14 x y+3 y^2-50 x+50 y+75=0$

Vedclass · Answer

(C) The given equation is $8 x^2-14 x y+3 y^2+10 x+10 y-25=0$.Factoring the quadratic part,we get $(4 x-y-5)(2 x-3 y+5)=0$.The lines are $L_1: 4 x-y-5=0$ and $L_2: 2 x-3 y+5=0$.The intersection point of $L_1$ and $L_2$ is $(2,3)$.Let the pair of lines $S=0$ be $(4 x-y+c_1)(2 x-3 y+c_2)=0$.Since the lines form a parallelogram,the diagonals intersect at the midpoint of the intersection points of the pairs of lines.Let the intersection of $S=0$ be $(x_1, y_1)$. The midpoint of $(2,3)$ and $(x_1, y_1)$ is $(3,2)$.Thus,$\frac{x_1+2}{2}=3 \Rightarrow x_1=4$ and $\frac{y_1+3}{2}=2 \Rightarrow y_1=1$.Substituting $(4,1)$ into $4 x-y+c_1=0$ gives $16-1+c_1=0 \Rightarrow c_1=-15$.Substituting $(4,1)$ into $2 x-3 y+c_2=0$ gives $8-3+c_2=0 \Rightarrow c_2=-5$.The equation $S=0$ is $(4 x-y-15)(2 x-3 y-5)=0$.Expanding this,we get $8 x^2-12 x y-20 x-2 x y+3 y^2+5 y-30 x+45 y+75=0$,which simplifies to $8 x^2-14 x y+3 y^2-50 x+50 y+75=0$.

$A$ pair of lines $S=0$ together with the lines given by the equation $8 x^2-14 x y+3 y^2+10 x+10 y-25=0$ form a parallelogram. If its diagonals intersect at the point $(3,2)$,then the equation $S=0$ is

Similar Questions

One line of the pair of lines $x^2+xy-2y^2=0$ is perpendicular to one line of the pair of lines $3y^2-5xy-2x^2=0$. If the combined equation of the two lines other than those two perpendicular lines is $ax^2+2hxy+by^2=0$,then $a+2h+b=$

If the pair of lines represented by $3x^2 - 5xy + Py^2 = 0$ and $6x^2 - xy - 5y^2 = 0$ have one line in common,then the sum of all possible values of $P$ is

The area bounded by the angle bisectors of the lines ${x^2} - {y^2} + 2y = 1$ and the line $x + y = 3$ is:

If the angle between the pair of lines $x^2+2 \sqrt{2} x y+k y^2=0, k>0$ is $45^{\circ}$,then the area (in square units) of the triangle formed by the pair of bisectors of angles between the given lines and the line $x+2 y+1=0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module