If x^2-y^2+2hxy+2gx+2fy+c=0 is the locus of a | vedclass

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

Question

(A) The locus of a point $P(x, y)$ equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$ is given by the angle bisectors of these lines.Equating the di

Vedclass · Accepted Answer

$14$

Vedclass · Answer

(A) The locus of a point $P(x, y)$ equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$ is given by the angle bisectors of these lines.Equating the distances:$\frac{|x+2y+7|}{\sqrt{1^2+2^2}} = \frac{|2x-y+8|}{\sqrt{2^2+(-1)^2}}$$|x+2y+7| = |2x-y+8|$$(x+2y+7)^2 - (2x-y+8)^2 = 0$$(x+2y+7 - (2x-y+8))(x+2y+7 + 2x-y+8) = 0$$(-x+3y-1)(3x+y+15) = 0$$(x-3y+1)(3x+y+15) = 0$$3x^2 + xy + 15x - 9xy - 3y^2 - 45y + 3x + y + 15 = 0$$3x^2 - 3y^2 - 8xy + 18x - 44y + 15 = 0$Dividing by $3$ to match the form $x^2-y^2+2hxy+2gx+2fy+c=0$:$x^2 - y^2 - \frac{8}{3}xy + 6x - \frac{44}{3}y + 5 = 0$Comparing coefficients:$2h = -\frac{8}{3} \implies h = -\frac{4}{3}$$2g = 6 \implies g = 3$$2f = -\frac{44}{3} \implies f = -\frac{22}{3}$$c = 5$Calculating $g+c+h-f = 3 + 5 + (-\frac{4}{3}) - (-\frac{22}{3}) = 8 - \frac{4}{3} + \frac{22}{3} = 8 + \frac{18}{3} = 8 + 6 = 14$.

If $x^2-y^2+2hxy+2gx+2fy+c=0$ is the locus of a point which moves such that it is always equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$,then the value of $g+c+h-f$ equals

Similar Questions

The equation of the pair of lines passing through the origin and forming an equilateral triangle with the line $3x + 4y - 5 = 0$ is

For what value of $m$ can the equation $y^2 + 2xy + 2x + my - 3 = 0$ be factored into two linear factors?

The area of the triangle formed by the lines $3x^2 - 4xy + y^2 = 0$ and $2x - y = 6$ is:

The equation of the pair of straight lines passing through the point $(1,1)$ and perpendicular to the pair of straight lines $3x^2-8xy+5y^2=0$ is

The equation of the image of the pair of lines $y = |x - 1|$ in the $y$-axis is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module