Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Show that the pair of straight lines 4x2 + 12xy + 9y2 – 6x – 9y + 2 = 0 represents two parallel straight lines and also find the separate equations of the straight lines.
If the lines 2x – 3y – 5 = 0 and 3x – 4y – 7 = 0 are the diameters of a circle, then its centre is:
Find the combined equation of the straight lines whose separate equations are x − 2y − 3 = 0 and x + y + 5 = 0
Show that 2x2 + 3xy − 2y2 + 3x + y + 1 = 0 represents a pair of perpendicular lines
Prove that the equation to the straight lines through the origin, each of which makes an angle α with the straight line y = x is x2 – 2xy sec 2α + y2 = 0
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
The slope of one of the straight lines ax2 + 2hxy + by2 = 0 is twice that of the other, show that 8h2 = 9ab
For what values of k does the equation 12x2 + 2kxy + 2y2 +11x – 5y + 2 = 0 represent two straight lines
Prove that one of the straight lines given by ax2 + 2hxy + by2 = 0 will bisect the angle between the coordinate axes if (a + b)2 = 4h2
Prove that the straight lines joining the origin to the points of intersection of 3x2 + 5xy – 3y2 + 2x + 3y = 0 and 3x – 2y – 1 = 0 are at right angles
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 coordinates of the four vertices of a quadrilateral are (−2, 4), (−1, 2), (1, 2) and (2, 4) taken in order. The equation of the line passing through the vertex (−1, 2) and dividing the quadrilateral in the equal areas is
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 length of ⊥ from the origin to the line `x/3 - y/4` = 1 is
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 ______.
The pair of lines represented by 3ax2 + 5xy + (a2 – 2)y2 = 0 are perpendicular to each other for ______.
