Advertisements
Advertisements
Question
If three lines whose equations are y = m1x + c1, y = m2x + c2 and y = m3x + c3 are concurrent, then show that m1(c2 – c3) + m2 (c3 – c1) + m3 (c1 – c2) = 0.
Solution
The equations of the given lines are
y = m1x + c1 …(1)
y = m2x + c2 …(2)
y = m3x + c3 …(3)
On subtracting equation (1) from (2), we obtain
0 = (m2 − m1)x + (c2 − c1)
= (m1 − m2)x = c2 − c1
= `x = (c_2 - c_1)/(m_1 - m_2)`
On substituting this value of x in (1), we obtain
`y = m_1 ((c_2 - c_1)/(m_1 - m_2)) + c_1`
`y = ((m_1c_2 - m_1c_1)/(m_1 - m_2)) + c_1`
`y = (m_1c_2 - m_1c_1 + m_1c_1 - m_2c_1)/(m_1 - m_2)`
`y = (m_1c_2 - m_2c_1)/(m_1 - m_2)`
∴ `((c_2 - c_1)/(m_1 - m_2), (m_1c_2 - m_2c_1)/(m_1 - m_2))` is the point of intersection of lines (1) and (2).
It is given that lines (1), (2), and (3) are concurrent. Hence, the point of intersection of lines (1) and (2) will also satisfy equation (3).
= `(m_1c_2 - m_2c_1)/(m_1 - m_2) = m_3 ((c_2 - c_1)/(m_1 - m_1)) + c_3`
= `(m_1c_2 - m_2c_1)/(m_1 - m_2) = (m_3c_2 - m_3c_1 + c_3m_1 - c_3m_2)/(m_1 - m_2)`
= m1c2 - m2c1 - m3c2 + m3c1 - c3m1 + c3m2 = 0
= m1 (c2 - c3) + m2 (c3 - c1) + m3 (c1 - c2) = 0
Hence, m1 (c2 - c3) + m2 (c3 - c1) + m3 (c1 - c2) = 0
APPEARS IN
RELATED QUESTIONS
The perpendicular from the origin to the line y = mx + c meets it at the point (–1, 2). Find the values of m and c.
If p and q are the lengths of perpendiculars from the origin to the lines x cos θ – y sin θ = k cos 2θ and xsec θ+ y cosec θ = k, respectively, prove that p2 + 4q2 = k2.
If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b, then show that `1/p^2 = 1/a^2 + 1/b^2`.
Show that the equation of the line passing through the origin and making an angle θ with the line `y = mx + c " is " y/c = (m+- tan theta)/(1 +- m tan theta)`.
Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0.
Prove that the product of the lengths of the perpendiculars drawn from the points `(sqrt(a^2 - b^2), 0)` and `(-sqrta^2-b^2, 0)` to the line `x/a cos theta + y/b sin theta = 1` is `b^2`.
Find the equation of a line which is equidistant from the lines x = − 2 and x = 6.
Find the lines through the point (0, 2) making angles \[\frac{\pi}{3} \text { and } \frac{2\pi}{3}\] with the x-axis. Also, find the lines parallel to them cutting the y-axis at a distance of 2 units below the origin.
Find the equations of the diagonals of the square formed by the lines x = 0, y = 0, x = 1 and y =1.
Find the equation of a line for p = 8, α = 300°.
Find the equation of the line on which the length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of x-axis is 30°.
If the straight line through the point P (3, 4) makes an angle π/6 with the x-axis and meets the line 12x + 5y + 10 = 0 at Q, find the length PQ.
Reduce the equation \[\sqrt{3}\] x + y + 2 = 0 to the normal form and find p and α.
Reduce the following equation to the normal form and find p and α in y − 2 = 0.
Reduce the equation 3x − 2y + 6 = 0 to the intercept form and find the x and y intercepts.
Find the coordinates of the vertices of a triangle, the equations of whose sides are
y (t1 + t2) = 2x + 2a t1t2, y (t2 + t3) = 2x + 2a t2t3 and, y (t3 + t1) = 2x + 2a t1t3.
Find the area of the triangle formed by the line y = 0, x = 2 and x + 2y = 3.
Find the equations of the medians of a triangle, the equations of whose sides are:
3x + 2y + 6 = 0, 2x − 5y + 4 = 0 and x − 3y − 6 = 0
Find the equation of the line joining the point (3, 5) to the point of intersection of the lines 4x + y − 1 = 0 and 7x − 3y − 35 = 0.
Prove that the following sets of three lines are concurrent:
3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 0
Find the conditions that the straight lines y = m1 x + c1, y = m2 x + c2 and y = m3 x + c3 may meet in a point.
Show that the straight lines L1 = (b + c) x + ay + 1 = 0, L2 = (c + a) x + by + 1 = 0 and L3 = (a + b) x + cy + 1 = 0 are concurrent.
Find the equation of the straight line perpendicular to 2x − 3y = 5 and cutting off an intercept 1 on the positive direction of the x-axis.
Find the equation of the straight line which has y-intercept equal to \[\frac{4}{3}\] and is perpendicular to 3x − 4y + 11 = 0.
Find the image of the point (2, 1) with respect to the line mirror x + y − 5 = 0.
If the image of the point (2, 1) with respect to the line mirror be (5, 2), find the equation of the mirror.
The equations of perpendicular bisectors of the sides AB and AC of a triangle ABC are x − y + 5 = 0 and x + 2y = 0 respectively. If the point A is (1, −2), find the equation of the line BC.
Write the coordinates of the orthocentre of the triangle formed by the lines x2 − y2 = 0 and x + 6y = 18.
Write the coordinates of the orthocentre of the triangle formed by the lines xy = 0 and x + y = 1.
The number of real values of λ for which the lines x − 2y + 3 = 0, λx + 3y + 1 = 0 and 4x − λy + 2 = 0 are concurrent is
Two vertices of a triangle are (−2, −1) and (3, 2) and third vertex lies on the line x + y = 5. If the area of the triangle is 4 square units, then the third vertex is
Find the equation of the lines which passes through the point (3, 4) and cuts off intercepts from the coordinate axes such that their sum is 14.
For specifying a straight line, how many geometrical parameters should be known?
Reduce the following equation into slope-intercept form and find their slopes and the y-intercepts.
6x + 3y – 5 = 0
Reduce the following equation into intercept form and find their intercepts on the axes.
3x + 2y – 12 = 0
Reduce the following equation into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
y − 2 = 0
