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
Advertisements
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
Find the angle between the pair of straight lines 3x2 – 5xy – 2y2 + 17x + y + 10 = 0.
If m1 and m2 are the slopes of the pair of lines given by ax2 + 2hxy + by2 = 0, then the value of m1 + m2 is:
If the lines 2x – 3y – 5 = 0 and 3x – 4y – 7 = 0 are the diameters of a circle, then its centre is:
Combined equation of co-ordinate axes is:
ax2 + 4xy + 2y2 = 0 represents a pair of parallel lines then ‘a’ is:
Show that 2x2 + 3xy − 2y2 + 3x + y + 1 = 0 represents a pair of perpendicular lines
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 separate equation of the following pair of straight lines
6(x – 1)2 + 5(x – 1)(y – 2) – 4(y – 3)2 = 0
The slope of one of the straight lines ax2 + 2hxy + by2 = 0 is twice that of the other, show that 8h2 = 9ab
Find p and q, if the following equation represents a pair of perpendicular lines
6x2 + 5xy – py2 + 7x + qy – 5 = 0
Show that the equation 9x2 – 24xy + 16y2 – 12x + 16y – 12 = 0 represents a pair of parallel lines. Find the distance between them
If the pair of straight lines x2 – 2kxy – y2 = 0 bisect the angle between the pair of straight lines x2 – 2lxy – y2 = 0, Show that the later pair also bisects the angle between the former
Choose the correct alternative:
If the equation of the base opposite to the vertex (2, 3) of an equilateral triangle is x + y = 2, then the length of a side is
Choose the correct alternative:
The image of the point (2, 3) in the line y = −x is
Choose the correct alternative:
The area of the triangle formed by the lines x2 – 4y2 = 0 and x = a is
If `"z"^2/(("z" - 1))` is always real, then z, can lie on ______.
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 ______.
The pair of lines represented by 3ax2 + 5xy + (a2 – 2)y2 = 0 are perpendicular to each other for ______.
