Question

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 mid point of the line AB.
Help pls :(​

Answer 1

Step-by-step explanation:

The line 2x+5y=1 meets the curve x^2+5xy-4y^2+10=0 at the points A and B. What are the coordinates of A and B?

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2x+5y=1. , or. y=(1–2x)/5………….(1)

x^2+5xy-4y^2+10=0…………………..(2)

Putting y=(1–2x)/5 from eqn. (1)

x^2+5x.(1–2x)/5–4/25.(1–2x)^2 +10=0

or. 25x^2+25x.(1–2x)-4.(1+4x^2–4x)+250 = 0

or. 25x^2+25x-50x^2–4–16x^2+16x+250=0

or. 41x^2–41x-246=0

or x^2– x - 6=0

or. (x-3).(x+2)=0

x= 3 or -2

But y=(1–2x)/5

y= (1–6)/5 or. (1+4)/5

y= -1. or. 1

Thus , A(3,-1) and B( -2,1). 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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