Advertisements
Advertisements
Question
Prove that the following sets of three lines are concurrent:
3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 0
Solution
Given:
3x − 5y − 11 = 0 ... (1)
5x + 3y − 7 = 0 ... (2)
x + 2y = 0 ... (3)
Now, consider the following determinant:
\[\begin{vmatrix}3 & - 5 & - 11 \\ 5 & 3 & - 7 \\ 1 & 2 & 0\end{vmatrix} = 3 \times 14 + 5 \times 7 - 11 \times 7 = 0\]
Hence, the given lines are concurrent.
APPEARS IN
RELATED QUESTIONS
Reduce the following equation into intercept form and find their intercepts on the axes.
3y + 2 = 0
Find angles between the lines `sqrt3x + y = 1 and x + sqrt3y = 1`.
In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.
Find the equation of a line for p = 8, α = 300°.
Find the value 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 equation of a straight line on which the perpendicular from the origin makes an angle of 30° with x-axis and which forms a triangle of area \[50/\sqrt{3}\] with the axes.
Reduce the equation \[\sqrt{3}\] x + y + 2 = 0 to slope-intercept form and find slope and y-intercept;
Reduce the equation\[\sqrt{3}\] x + y + 2 = 0 to intercept form and find intercept on the axes.
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 \[x + \sqrt{3}y - 4 = 0\] .
Reduce the following equation to the normal form and find p and α in \[x - y + 2\sqrt{2} = 0\].
Reduce the following equation to the normal form and find p and α in x − 3 = 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.
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.
If the three lines ax + a2y + 1 = 0, bx + b2y + 1 = 0 and cx + c2y + 1 = 0 are concurrent, show that at least two of three constants a, b, c are equal.
Find the equation of the perpendicular bisector of the line joining the points (1, 3) and (3, 1).
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 projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).
Find the values of α so that the point P (α2, α) lies inside or on the triangle formed by the lines x − 5y+ 6 = 0, x − 3y + 2 = 0 and x − 2y − 3 = 0.
Write the area of the figure formed by the lines a |x| + b |y| + c = 0.
The point which divides the join of (1, 2) and (3, 4) externally in the ratio 1 : 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
The equations of the sides AB, BC and CA of ∆ ABC are y − x = 2, x + 2y = 1 and 3x + y + 5 = 0 respectively. The equation of the altitude through B is
The figure formed by the lines ax ± by ± c = 0 is
The centroid of a triangle is (2, 7) and two of its vertices are (4, 8) and (−2, 6). The third vertex is
If the lines x + q = 0, y − 2 = 0 and 3x + 2y + 5 = 0 are concurrent, then the value of q will be
A point equidistant from the line 4x + 3y + 10 = 0, 5x − 12y + 26 = 0 and 7x+ 24y − 50 = 0 is
Find the equation of a line which passes through the point (2, 3) and makes an angle of 30° with the positive direction of x-axis.
If the intercept of a line between the coordinate axes is divided by the point (–5, 4) in the ratio 1 : 2, then find the equation of the line.
A line cutting off intercept – 3 from the y-axis and the tangent at angle to the x-axis is `3/5`, its equation is ______.
For specifying a straight line, how many geometrical parameters should be known?
A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is ______.
The line which cuts off equal intercept from the axes and pass through the point (1, –2) 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
