Question

The curve y = x^2-4x-5 and the line y = 1 - 3x meet at the point a and
b. Find the coordiantes of a and b

Answer 1

Question:

The curve y = x^2 - 4x - 5 and the line

y = 1 - 3x meet at the points A and B .

Find the coordinates of points A and B.

Answer:

Coordinate of points A and B are

(-2,7) and (3,-8) respectively.

Given:

The curve y = x^2 - 4x - 5 and the line

y = 1 - 3x meet at the points A and B .

To find:

Coordinates of points A and B.

Solution:

Let;

y = x^2 - 4x - 5 --------(1)

y = 1 - 3x ---------(2)

It is given that , the curve and the line meet (intersect) each other.

Thus, from eq-(1) and eq-(2) , we have;

=> 1 - 3x = x^2 - 4x - 5

=> x^2 - 4x - 5 + 3x - 1 = 0

=> x^2 - x - 6 = 0

=> x^2 - 3x + 2x - 6 = 0

=> x(x - 3) + 2(x - 3) = 0

=> (x - 3)(x + 2) = 0

=> x = -2 , 3

Case(1) ,

When x = -2 ,then using eq-(2) ,we have;

=> y = 1 - 3x

=> y = 1 - 3•(-2)

=> y = 1 + 6

=> y = 7

Case(2)

When x = 3 ,then using eq-(2) ,we have;

=> y = 1 - 3x

=> y = 1 - 3•3

=> y = 1 - 9

=> y = - 8

Thus,

The required points of intersection are

(-2,7) and (3,-8).

Hence,

The curve and the line intersect each other at the points A(-2,7) and B(3,-8).

Answer 2

Answer:

$\bold\red{a ( -2, 7 ) }\: \\\\ \:\bold\red{b (3, -8 )}$

Step-by-step explanation:

Given,

the curve

$y = {x}^{2} - 4x - 5$

and

the line

$y = 1 - 3x$

meet at the point 'a' and 'b'

Since the line meets the curve,

therefore,

it will satisfy the equation of the curve also at these two points

for this,

let's put the value of 'y' from equation of line in the equation of curve

therefore,

we get,

$= > 1 - 3x = {x}^{2} - 4x - 5 \\ \\ = > {x}^{2} - 4x + 3x - 5 - 1 = 0 \\ \\ = > {x}^{2} - x - 6 = 0 \\ \\ = > {x}^{2} - 3x + 2x - 6 = 0 \\ \\ = > x(x - 3) + 2(x - 3) = 0 \\ \\ = > (x + 2)(x - 3) = 0 \\ \\ = > x = - 2 \\ \\ and \\ \\ = > x = 3$

Therefore,

$= > y = 1 - 3 \times ( - 2) = 1 + 6 = 7 \\ \\ and \\ \\ = > y = 1 - 3 \times 3 = 1 - 9 = - 8$

Hence,

the points where the line and curve meet are

a ( -2, 7 ) and b (3, -8 )