If the equation ax^2+2hxy+by^2+2gx+2fy+c=0 re | vedclass

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

Question

(C) The point of intersection $(x_0, y_0)$ of the pair of straight lines $ax^2+2hxy+by^2+2gx+2fy+c=0$ is given by the solution of the partial derivati

Vedclass · Accepted Answer

$\frac{c(a+b)-f^2-g^2}{ab-h^2}$

Vedclass · Answer

(C) The point of intersection $(x_0, y_0)$ of the pair of straight lines $ax^2+2hxy+by^2+2gx+2fy+c=0$ is given by the solution of the partial derivatives $\frac{\partial}{\partial x} = 0$ and $\frac{\partial}{\partial y} = 0$. $2ax+2hy+2g=0 \implies ax+hy+g=0$ $2hx+2by+2f=0 \implies hx+by+f=0$ Solving these equations using Cramer's rule: $x_0 = \frac{hf-bg}{ab-h^2}$ and $y_0 = \frac{gh-af}{ab-h^2}$. The square of the distance from the origin $(0,0)$ is $D^2 = x_0^2 + y_0^2$. Using the condition for a pair of lines $abc+2fgh-af^2-bg^2-ch^2=0$,we can simplify the expression for $x_0^2+y_0^2$. Substituting the values,we get $D^2 = \frac{(hf-bg)^2+(gh-af)^2}{(ab-h^2)^2} = \frac{h^2f^2+b^2g^2-2hfbg+g^2h^2+a^2f^2-2ghaf}{(ab-h^2)^2}$. Using $abc+2fgh=af^2+bg^2+ch^2$,this simplifies to $\frac{c(a+b)-f^2-g^2}{ab-h^2}$.

If the equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of straight lines,then the square of the distance of their point of intersection from the origin is

Similar Questions

The joint equation of the pair of lines which bisects the angles between the lines $x^2+3xy+2y^2=0$ is

If the pair of straight lines $x^2 - 2mxy - y^2 = 0$ and $x^2 - 2nxy - y^2 = 0$ are such that each pair bisects the angle between the other pair,then $mn =$

The locus of the points equidistant from the lines represented by $x^2 \cos^2 \theta - xy \sin^2 \theta - y^2 \sin^2 \theta = 0$ is

The area of the triangle formed by the line $x+y=4$ and the angle bisectors of the pair of lines $x^2-y^2+2y-1=0$ is ......... sq. units.

If the angle between the pair of lines $2x^2 + 2hxy + 2y^2 - x + y - 1 = 0$ is $\tan^{-1}(3/4)$ and $h$ is a positive rational number,then the point of intersection of these two lines is

Vedclass Test Series

Exam Paper Generator

Online Exam Module