Question

1
A curve has equation y = 2xy + 5 and a line has equation 2x + 5y = 1.
The curve and the line intersect at the points A and B. Find the coordinates of the midpoint
of the line AB.

Answer 1

Answer:

here is your answer....☺️

Answer 2

Step-by-step explanation:

Given:

A curve y=2xy+5 and a line 2x+5y=1

To find:The curve and the line intersect at the points A and B. Find the coordinates of the midpoint of the line AB.

Solution:

Step 1:Put the value of y from line into curve

$5y = 1 - 2x \\ \\ y = \frac{1 - 2x}{5} ...eq1$

$\frac{1 - 2x}{5} = 2x \left(\frac{1 - 2x}{5} \right) + 5$

Step 2: Take LCM in RHS and cancel 5 from both sides

$1 - 2x = 2x(1 - 2x) + 25 \\ \\ 1 - 2x = 2x - 4 {x}^{2} + 25 \\ \\ or \\ \\ 4 {x}^{2} - 4x - 24 = 0 \\ \\ or \\ \\ 4( {x}^{2} - x - 6) = 0$

Step 3: Find the roots of quadratic equation

${x}^{2} - x - 6 = 0 \\ \\ {x}^{2} - 3x + 2x - 6 = 0 \\ \\ x(x - 3) + 2(x - 3) = 0 \\ \\ (x - 3)(x + 2) = 0 \\ \\ x = 3 \\ \\ or \\ \\ x = - 2$

Step 4:Put value of x in eq1 and find value of y

when x=3

$y = \frac{1 - 2 \times 3}{5} \\ \\ y = \frac{1 - 6}{5} \\ \\ y = - 1$

Let this is point A(3,-1)

Put x=-2

$y = \frac{1 - 2( - 2)}{5} \\ \\ y = \frac{1 + 4}{5} \\ \\ y = 1$

Let this is point B(-2,1)

Step 5: Find mid-point of line segment AB.

Let mid-point of AB is C.

Find coordinates of C(x,y) using mid-point formula.

$x = \frac{ 3 - 2}{2} \\ \\ x = \frac{1}{2} \\ \\ x = 0.5 \\ \\ y = \frac{1 - 1}{2} \\ \\ y = \frac{0}{2} \\ \\ y = 0$

Coordinates of mid-point C(0.5,0)

Final answer:

Coordinates of A(3,-1) and B(-2,1) and C(0.5,0).

Hope it helps you.

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