The lines joining ihe origin to the points of intersection of 2x2 + 3xy - 4x + 1 = 0 and 3x + y - 1 = 0 are given bya)x2+y2-5xy=0b)x2-y2-5xy=0c)x2+y2+5xy=0d)x2-y2+5xy=0Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Given Equations:
- 2x^2 + 3xy - 4x + 1 = 0
- 3x - y - 1 = 0

Step 1: Find the points of intersection
To find the points of intersection, we need to solve the given system of equations simultaneously.

Step 2: Solve the system of equations
Using substitution method, we can solve the given system of equations as follows:

From equation (2), we can express y in terms of x:
y = 3x - 1

Substituting this value of y in equation (1), we get:
2x^2 + 3x(3x - 1) - 4x + 1 = 0
2x^2 + 9x^2 - 3x - 4x + 1 = 0
11x^2 - 7x + 1 = 0

Step 3: Find the points of intersection
To find the points of intersection, we need to solve the quadratic equation 11x^2 - 7x + 1 = 0.

Using the quadratic formula, we get:
x = (-(-7) ± √((-7)^2 - 4(11)(1))) / (2(11))
x = (7 ± √(49 - 44)) / 22
x = (7 ± √5) / 22

Substituting the values of x in equation (2), we can find the corresponding values of y.

When x = (7 + √5) / 22:
y = 3(7 + √5) / 22 - 1
y = (21 + 3√5 - 22) / 22
y = (3√5 - 1) / 22

When x = (7 - √5) / 22:
y = 3(7 - √5) / 22 - 1
y = (21 - 3√5 - 22) / 22
y = (-3√5 - 1) / 22

Therefore, the points of intersection are:
A = ( (7 + √5) / 22, (3√5 - 1) / 22 )
B = ( (7 - √5) / 22, (-3√5 - 1) / 22 )

Step 4: Find the equation of the line passing through the origin and these points of intersection

The equation of a line passing through two points (x1, y1) and (x2, y2) is given by:
(y - y1) / (y2 - y1) = (x - x1) / (x2 - x1)

Using point A = ( (7 + √5) / 22, (3√5 - 1) / 22 ) and the origin (0, 0), we can find the equation of the line passing through these two points.

(y - 0) / ((3√5 - 1) / 22 - 0) = (x - 0) / ((7 + √5) /

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