Question Statement Find the length of the chord intercepted by the line 2 x + y − 1 = 0 2 x 2 − 3 y 2 + 7 x − 4 y + 13 = 0
Background and Explanation To find the length of a chord formed by a line intersecting a conic section, we solve the equations simultaneously to find the coordinates of the intersection points. Once the two points ( x 1 , y 1 ) ( x 2 , y 2 )
Solution Given the equation of the line:
2 x + y − 1 = 0
And the equation of the hyperbola:
2 x 2 − 3 y 2 + 7 x − 4 y + 13 = 0
From equation (1), we can isolate y y = − 2 x + 1
Substitute equation (3) into equation (2):
2 x 2 − 3 ( − 2 x + 1 ) 2 + 7 x − 4 ( − 2 x + 1 ) + 13 = 0
Expand the squared term and distribute the constants:
2 x 2 − 3 ( 4 x 2 − 4 x + 1 ) + 7 x + 8 x − 4 + 13 = 0 2 x 2 − 12 x 2 + 12 x − 3 + 7 x + 8 x − 4 + 13 = 0
Combine like terms:
− 10 x 2 + 27 x + 6 = 0
Multiply the entire equation by − 1 10 x 2 − 27 x − 6 = 0
Using the quadratic formula x = 2 a − b ± b 2 − 4 a c x = 2 × 10 − ( − 27 ) ± ( − 27 ) 2 − 4 ( 10 ) ( − 6 ) x = 20 27 ± 729 + 240 x = 20 27 ± 969
The two x x 1 = 20 27 + 969 , x 2 = 20 27 − 969
Substitute x 1 x 2
For x 1
y 1 = − 2 ( 20 27 + 969 ) + 1 y 1 = 10 − 27 − 969 + 10 10 = 10 − 17 − 969
For x 2
y 2 = − 2 ( 20 27 − 969 ) + 1 y 2 = 10 − 27 + 969 + 10 10 = 10 − 17 + 969
The endpoints of the chord are:
A ( 20 27 + 969 , 10 − 17 − 969 ) and B ( 20 27 − 969 , 10 − 17 + 969 )
Using the distance formula d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Δ x = 20 27 − 969 − ( 27 + 969 ) = 20 − 2 969 = 10 − 969 Δ y = 10 − 17 + 969 − ( − 17 − 969 ) = 10 2 969
Now, calculate the distance:
∣ A B ∣ = ( 10 − 969 ) 2 + ( 10 2 969 ) 2 ∣ A B ∣ = 100 969 + 100 4 ( 969 ) ∣ A B ∣ = 100 969 + 3876 = 100 4845 ∣ A B ∣ = 10 4845
Substitution Method: Used to solve the system of linear and quadratic equations.Quadratic Formula: x = 2 a − b ± b 2 − 4 a c Distance Formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Summary of Steps
Isolate y 2 x + y − 1 = 0
Substitute the expression for y x
Solve the quadratic equation 10 x 2 − 27 x − 6 = 0
Find the two intersection points ( x 1 , y 1 ) ( x 2 , y 2 ) x
Apply the distance formula between the two points to find the final chord length 10 4845