Advertisements
Advertisements
प्रश्न
Prove that the following sets of three lines are concurrent:
\[\frac{x}{a} + \frac{y}{b} = 1, \frac{x}{b} + \frac{y}{a} = 1\text { and } y = x .\]
उत्तर
Given:
\[bx + ay - ab = 0\] ... (1)
\[ax + by - ab = 0\] ... (2)
x − y = 0 ... (3)
Now, consider the following determinant:
\[\begin{vmatrix}b & a & - ab \\ a & b & - ab \\ 1 & - 1 & 0\end{vmatrix} = - b \times ab - a \times ab - ab \times \left( - a - b \right) = 0\]
Hence, the given lines are concurrent.
APPEARS IN
संबंधित प्रश्न
Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.
Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and –6, respectively.
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.
In what ratio, the line joining (–1, 1) and (5, 7) is divided by the line x + y = 4?
Find equation of the line which is equidistant from parallel lines 9x + 6y – 7 = 0 and 3x + 2y + 6 = 0.
A person standing at the junction (crossing) of two straight paths represented by the equations 2x – 3y+ 4 = 0 and 3x + 4y – 5 = 0 wants to reach the path whose equation is 6x – 7y + 8 = 0 in the least time. Find equation of the path that he should follow.
Find the equation of a line making an angle of 150° with the x-axis and cutting off an intercept 2 from y-axis.
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 equation of the line which intercepts a length 2 on the positive direction of the x-axis and is inclined at an angle of 135° with the positive direction of y-axis.
Find the equations of the sides of the triangles the coordinates of whose angular point is respectively (1, 4), (2, −3) and (−1, −2).
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°.
The length of the perpendicular from the origin to a line is 7 and the line makes an angle of 150° with the positive direction of Y-axis. Find the equation of the line.
Reduce the equation \[\sqrt{3}\] x + y + 2 = 0 to slope-intercept form and find slope and y-intercept;
Reduce the following equation to the normal form and find p and α in x − 3 = 0.
Put the equation \[\frac{x}{a} + \frac{y}{b} = 1\] to the slope intercept form and find its slope and y-intercept.
Find the values of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line \[\sqrt{3}x + y + 2 = 0\].
Find the coordinates of the vertices of a triangle, the equations of whose sides are x + y − 4 = 0, 2x − y + 3 = 0 and x − 3y + 2 = 0.
Show that the area of the triangle formed by the lines y = m1 x, y = m2 x and y = c is equal to \[\frac{c^2}{4}\left( \sqrt{33} + \sqrt{11} \right),\] where m1, m2 are the roots of the equation \[x^2 + \left( \sqrt{3} + 2 \right)x + \sqrt{3} - 1 = 0 .\]
Find the orthocentre of the triangle the equations of whose sides are x + y = 1, 2x + 3y = 6 and 4x − y + 4 = 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.
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 values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines x + y − 4 = 0, 3x − 7y − 8 = 0 and 4x − y − 31 = 0.
Determine whether the point (−3, 2) lies inside or outside the triangle whose sides are given by the equations x + y − 4 = 0, 3x − 7y + 8 = 0, 4x − y − 31 = 0 .
If a ≠ b ≠ c, write the condition for which the equations (b − c) x + (c − a) y + (a − b) = 0 and (b3 − c3) x + (c3 − a3) y + (a3 − b3) = 0 represent the same line.
The figure formed by the lines ax ± by ± c = 0 is
A point equidistant from the line 4x + 3y + 10 = 0, 5x − 12y + 26 = 0 and 7x+ 24y − 50 = 0 is
A line passes through P(1, 2) such that its intercept between the axes is bisected at P. The equation of the line 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 what values of a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0 are equal in length but opposite in signs to those cut off by the line 2x – 3y + 6 = 0 on the axes.
If the line `x/"a" + y/"b"` = 1 passes through the points (2, –3) and (4, –5), then (a, b) is ______.
If the coordinates of the middle point of the portion of a line intercepted between the coordinate axes is (3, 2), then the equation of the line will be ______.
A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is ______.
Locus of the mid-points of the portion of the line x sin θ + y cos θ = p intercepted between the axes is ______.
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 slope-intercept form and find their slopes and the y-intercepts.
y = 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
