Advertisements
Advertisements
प्रश्न
Prove that the following sets of three lines are concurrent:
15x − 18y + 1 = 0, 12x + 10y − 3 = 0 and 6x + 66y − 11 = 0
उत्तर
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
संबंधित प्रश्न
Reduce the following equation into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
`x – sqrt3y + 8 = 0`
Two lines passing through the point (2, 3) intersects each other at an angle of 60°. If slope of one line is 2, find equation of the other line.
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`.
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.
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.
Point R (h, k) divides a line segment between the axes in the ratio 1 : 2. Find the equation of the line.
Find the equation of a line for p = 5, α = 60°.
Find the equation of a line for p = 8, α = 225°.
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°.
Reduce the equation\[\sqrt{3}\] x + y + 2 = 0 to intercept form and find intercept on the axes.
Reduce the following equation to the normal form and find p and α in \[x + y + \sqrt{2} = 0\].
Reduce the following equation to the normal form and find p and α in x − 3 = 0.
Reduce the following equation to the normal form and find p and α in y − 2 = 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 point of intersection of the following pairs of lines:
bx + ay = ab and ax + by = ab.
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.
Prove that the following sets of three lines are concurrent:
3x − 5y − 11 = 0, 5x + 3y − 7 = 0 and x + 2y = 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 lines p1 x + q1 y = 1, p2 x + q2 y = 1 and p3 x + q3 y = 1 be concurrent, show that the points (p1, q1), (p2, q2) and (p3, q3) are collinear.
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 a line which is perpendicular to the line \[\sqrt{3}x - y + 5 = 0\] and which cuts off an intercept of 4 units with the negative direction of y-axis.
Find the coordinates of the foot of the perpendicular from the point (−1, 3) to the line 3x − 4y − 16 = 0.
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.
Write the coordinates of the orthocentre of the triangle formed by the lines x2 − y2 = 0 and x + 6y = 18.
Write the area of the figure formed by the lines a |x| + b |y| + c = 0.
A (6, 3), B (−3, 5), C (4, −2) and D (x, 3x) are four points. If ∆ DBC : ∆ ABC = 1 : 2, then x is equal to
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
If the lines x + q = 0, y − 2 = 0 and 3x + 2y + 5 = 0 are concurrent, then the value of q will be
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.
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.
y = 0
Reduce the following equation into intercept form and find their intercepts on the axes.
4x – 3y = 6
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
