Question

2. If a : b: c=1: 2: 3 then the lines represented by ax²+bxy+cy = 0 are​

Answer 1

Step-by-step explanation:

If a : b: c=1: 2: 3 then the lines represented by ax²+bxy+cy = 0 are

Both are conic sections: they can be obtained as the intersection of a cone and a plane. This results in the general expression of points on them as satisfying the equation ax2+bxy+cy2+dx+ey+f=0; if we have b2−4ac>0 , this describes a hyperbola, and if b2−4ac=0 , it is a parabola. (If b2−4ac<0 , it's an ellipse.) As a particular example, cy2+dx=0 is a parabola if c,d≠0 , and ax2+cy2+f=0 is a hyperbola if ac<0 .

Answer 2

Given,

• a : b: c=1: 2: 3

• Equation of lines ax²+bxy+cy = 0

To find,

What do the given lines represent?

Solution,

The ratios can be multiplied by some constant of proportion to get the exact values. Let us say that k is the constant of proportion then,

a = k

b = 2k

c = 3k

Substituting their values in the given equation we have,

    kx²+2kxy+3ky = 0

    x²+2xy+3y = 0

Dividing the equation by $y^{2}$ and let $\frac{x}{y} = t$

   $t^{2} + 2t + 3 = 0$

Calculating the value of D, we obtain a negative in the square root part, indicating that the solutions for t are imaginary. On solving the given equation using the quadratic formula, we obtain the roots as follows:-

$\frac{x}{y} = -1 + \sqrt{2}i , -1 - \sqrt{2}i$

The equation doesn't represent actual lines on the cartesian plane.

The equation represents two lines intersecting at origin on the imaginary plane, forming a continuous 'X' like the graph.