The lines joining the points of intersection | vedclass

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

Question

(C) To find the lines joining the origin to the points of intersection,we homogenize the equation of the curve using the line equation $x - y = 2$,whi

Vedclass · Accepted Answer

Equal

Vedclass · Answer

(C) To find the lines joining the origin to the points of intersection,we homogenize the equation of the curve using the line equation $x - y = 2$,which implies $\frac{x - y}{2} = 1$.The equation of the curve is $5x^2 + 12xy - 8y^2 + 8x - 4y + 12 = 0$.Substituting $1$ with $\frac{x - y}{2}$ in the linear and constant terms:$5x^2 + 12xy - 8y^2 + (8x - 4y)\left(\frac{x - y}{2}\right) + 12\left(\frac{x - y}{2}\right)^2 = 0$Expanding the terms:$5x^2 + 12xy - 8y^2 + (4x^2 - 4xy - 2xy + 2y^2) + 12\left(\frac{x^2 - 2xy + y^2}{4}\right) = 0$$5x^2 + 12xy - 8y^2 + 4x^2 - 6xy + 2y^2 + 3(x^2 - 2xy + y^2) = 0$$5x^2 + 12xy - 8y^2 + 4x^2 - 6xy + 2y^2 + 3x^2 - 6xy + 3y^2 = 0$Combining like terms:$(5 + 4 + 3)x^2 + (12 - 6 - 6)xy + (-8 + 2 + 3)y^2 = 0$$12x^2 + 0xy - 3y^2 = 0$$12x^2 - 3y^2 = 0$$4x^2 - y^2 = 0$This factors as $(2x - y)(2x + y) = 0$,so the lines are $y = 2x$ and $y = -2x$.Since the slopes are $m_1 = 2$ and $m_2 = -2$,the lines are symmetric with respect to the axes. The angles made with the axes are equal in magnitude.

The lines joining the points of intersection of the curve $5x^2 + 12xy - 8y^2 + 8x - 4y + 12 = 0$ and the line $x - y = 2$ to the origin make angles with the axes that are:

Similar Questions

If the angle between the lines joining the origin to the points of intersection of $x+2y+\lambda=0$ and $2x^2-2xy+3y^2+2x-y-1=0$ is $\frac{\pi}{2}$,then a value of $\lambda$ is

If the lines joining the origin to the points of intersection of the curve $2x^2 - 2xy + 3y^2 + 2x - y - 1 = 0$ and the line $x + 2y = k$ are at right angles,then $k^2$ equals

The lines represented by the equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ will be equidistant from the origin,if

If $\theta$ is the angle between the lines joining the origin to the points of intersection of the curve $2x^2 + 3y^2 = 6$ and the line $x + y = 1$,then $\sin \theta =$

The acute angle formed between the lines joining the origin to the points of intersection of the curves $x^2 + y^2 - 2x - 1 = 0$ and $x + y = 1$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module