Advertisements
Advertisements
Question
Prove that the following sets of three lines are concurrent:
15x − 18y + 1 = 0, 12x + 10y − 3 = 0 and 6x + 66y − 11 = 0
Solution
Given:
15x − 18y + 1 = 0 ... (1)
12x + 10y − 3 = 0 ... (2)
6x + 66y − 11 = 0 ... (3)
Now, consider the following determinant:
\[\begin{vmatrix}15 & - 18 & 1 \\ 12 & 10 & - 3 \\ 6 & 66 & - 11\end{vmatrix} = 15\left( - 110 + 198 \right) + 18\left( - 132 + 18 \right) + 1\left( 792 - 60 \right)\]
\[\Rightarrow \begin{vmatrix}15 & - 18 & 1 \\ 12 & 10 & - 3 \\ 6 & 66 & - 11\end{vmatrix} = 1320 - 2052 + 732 = 0\]
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 equation of the line parallel to the line 3x – 4y + 2 = 0 and passing through the point (–2, 3).
Find angles between the lines `sqrt3x + y = 1 and x + sqrt3y = 1`.
Find the coordinates of the foot of perpendicular from the point (–1, 3) to the line 3x – 4y – 16 = 0.
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 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`.
Find the equation of the line passing through the point of intersection of the lines 4x + 7y – 3 = 0 and 2x – 3y + 1 = 0 that has equal intercepts on the axes.
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 making an angle of 150° with the x-axis and cutting off an intercept 2 from 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 the side BC of the triangle ABC whose vertices are (−1, −2), (0, 1) and (2, 0) respectively. Also, find the equation of the median through (−1, −2).
Find the equation of a line for p = 4, α = 150°.
Find the equation of a line for p = 8, α = 225°.
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 x + y − 6 = 0, x − 3y − 2 = 0 and 5x − 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 \[x^2 + \left( \sqrt{3} + 2 \right)x + \sqrt{3} - 1 = 0 .\]
For what value of λ are the three lines 2x − 5y + 3 = 0, 5x − 9y + λ = 0 and x − 2y + 1 = 0 concurrent?
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.
If a, b, c are in A.P., prove that the straight lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 and cx + 4y + 1 = 0 are concurrent.
Find the equation of the right bisector of the line segment joining the points (a, b) and (a1, b1).
If the image of the point (2, 1) with respect to the line mirror be (5, 2), find the equation of the mirror.
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 .
Write the coordinates of the orthocentre of the triangle formed by the lines x2 − y2 = 0 and x + 6y = 18.
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.
If the lines ax + 12y + 1 = 0, bx + 13y + 1 = 0 and cx + 14y + 1 = 0 are concurrent, then a, b, c are in
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.
Prove that every straight line has an equation of the form Ax + By + C = 0, where A, B and C are constants.
If the line `x/"a" + y/"b"` = 1 passes through the points (2, –3) and (4, –5), then (a, b) 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 normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
x − y = 4
