Advertisements
Advertisements
प्रश्न
A ∆OPQ is formed by the pair of straight lines x2 – 4xy + y2 = 0 and the line PQ. The equation of PQ is x + y – 2 = 0, Find the equation of the median of the triangle ∆ OPQ drawn from the origin O
उत्तर
The equation of the given pair of lines is
x2 – 4xy + y2 = 0 .......(1)
The equation of the line PQ is
x + y – 2 = 0
y = 2 – x .......(2)
To find the coordinates of P and Q, .
Solve equations (1) and (2)
(1) ⇒ x2 – 4x (2 – x) + (2 – x)2 = 0
x2 – 8x + 4x2 + 4 – 4x + x2 = 0
6x2 – 12x + 4 = 0
3x2 – 6x + 2 = 0
x = `(6 +- sqrt(6^2 - 4 xx 3 xx 2))/(2 xx 3)`
= `(6 +- sqrt(36 - 24))/6`
= `(6 +- sqrt(12))/6`
= `(6 +- 2sqrt(3))/6`
= `(3 +- sqrt(3))/3`
When x = `(3 +- sqrt(3))/3`, y = `2 - (3 +- sqrt(3))/3`
y = `(6 - 3 - sqrt(3))/3`
= `(3 - sqrt(3))/3`
When x = `(3 - sqrt(3))/3`, y = `2 - (3 - sqrt(3))/3`
y = `(6 - 3 + sqrt(3))/3`
= `(3 + sqrt(3))/3`
∴ P is `((3 + sqrt(3))/3, (3 - sqrt(3))/3)`
and
Q is `((3 - sqrt(3))/3, (3 + sqrt(3))/3)`
The midpoint of PQ is
D = `(((3 + sqrt(3))/3 + (3 - sqrt(3))/3)/3, ((3 - sqrt(3))/3 + (3 + sqrt(3))/3)/3)`
= `((3 + sqrt(3) + 3 - sqrt(3))/6, (3 - sqrt(3) + 3 + sqrt(3))/6)`
= `(6/6, 6/6)`
= (1, 1)
The equation of the median drawn from 0 is the equation of the line joining 0(0, 0) and D(1, 1)
`(x - 0)/(1 - 0) = (y - 0)/(1 - 0)`
⇒ `x/1 = y/1`
∴ The required equation is x = y
APPEARS IN
संबंधित प्रश्न
If the equation ax2 + 5xy – 6y2 + 12x + 5y + c = 0 represents a pair of perpendicular straight lines, find a and c.
Show that the equation 12x2 – 10xy + 2y2 + 14x – 5y + 2 = 0 represents a pair of straight lines and also find the separate equations of the straight lines.
Find the angle between the pair of straight lines 3x2 – 5xy – 2y2 + 17x + y + 10 = 0.
If the lines 2x – 3y – 5 = 0 and 3x – 4y – 7 = 0 are the diameters of a circle, then its centre is:
Show that the equation 2x2 − xy − 3y2 − 6x + 19y − 20 = 0 represents a pair of intersecting lines. Show further that the angle between them is tan−1(5)
Find the equation of the pair of straight lines passing through the point (1, 3) and perpendicular to the lines 2x − 3y + 1 = 0 and 5x + y − 3 = 0
Find the separate equation of the following pair of straight lines
3x2 + 2xy – y2 = 0
Find the separate equation of the following pair of straight lines
6(x – 1)2 + 5(x – 1)(y – 2) – 4(y – 3)2 = 0
Find the separate equation of the following pair of straight lines
2x2 – xy – 3y2 – 6x + 19y – 20 = 0
The slope of one of the straight lines ax2 + 2hxy + by2 = 0 is twice that of the other, show that 8h2 = 9ab
Find the value of k, if the following equation represents a pair of straight lines. Further, find whether these lines are parallel or intersecting, 12x2 + 7xy − 12y2 − x + 7y + k = 0
Show that the equation 9x2 – 24xy + 16y2 – 12x + 16y – 12 = 0 represents a pair of parallel lines. Find the distance between them
Choose the correct alternative:
Equation of the straight line that forms an isosceles triangle with coordinate axes in the I-quadrant with perimeter `4 + 2sqrt(2)` is
Choose the correct alternative:
The area of the triangle formed by the lines x2 – 4y2 = 0 and x = a is
Choose the correct alternative:
If one of the lines given by 6x2 – xy – 4cy2 = 0 is 3x + 4y = 0, then c equals to ______.
The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A'B (where B is the point (2, 3)) subtend angle `π/4` at the origin, is equal to ______.
Let the equation of the pair of lines, y = px and y = qx, can be written as (y – px) (y – qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 – 4xy – 5y2 = 0 is ______.
