Advertisements
Advertisements
Question
The equation of the line with slope −3/2 and which is concurrent with the lines 4x + 3y − 7 = 0 and 8x + 5y − 1 = 0 is
Options
3x + 2y − 63 = 0
3x + 2y − 2 = 0
2y − 3x − 2 = 0
none of these
Advertisements
Solution
3x + 2y − 2 = 0
Given:
4x + 3y − 7 = 0 ... (1)
8x + 5y − 1 = 0 ... (2)
The equation of the line with slope \[- \frac{3}{2}\] is given below: \[y = - \frac{3}{2}x + c\] \[\Rightarrow \frac{3}{2}x + y - c = 0\] ... (3)
The lines (1), (2) and (3) are concurrent.
\[\therefore \begin{vmatrix}4 & 3 & - 7 \\ 8 & 5 & - 1 \\ \frac{3}{2} & 1 & - c\end{vmatrix} = 0\]
\[ \Rightarrow 4\left( - 5c + 1 \right) - 3\left( - 8c + \frac{3}{2} \right) - 7\left( 8 - \frac{15}{2} \right) = 0\]
\[ \Rightarrow - 20c + 4 + 24c - \frac{9}{2} - 56 + \frac{105}{2} = 0\]
\[ \Rightarrow \frac{- 40c + 8 + 48c - 9 - 112 + 105}{2} = 0\]
\[ \Rightarrow 8c = 8\]
\[ \Rightarrow c = 1\]
On substituting c = 1 in \[y = - \frac{3}{2}x + c\], we get:
\[y = - \frac{3}{2}x + 1\]
\[ \Rightarrow 3x + 2y - 2 = 0\]
APPEARS IN
RELATED QUESTIONS
Draw a quadrilateral in the Cartesian plane, whose vertices are (–4, 5), (0, 7), (5, –5) and (–4, –2). Also, find its area.
Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
The slope of a line is double of the slope of another line. If tangent of the angle between them is `1/3`, find the slopes of the lines.
Find the slope of the lines which make the following angle with the positive direction of x-axis:
\[\frac{2\pi}{3}\]
State whether the two lines in each of the following is parallel, perpendicular or neither.
Through (6, 3) and (1, 1); through (−2, 5) and (2, −5)
Show that the line joining (2, −3) and (−5, 1) is parallel to the line joining (7, −1) and (0, 3).
By using the concept of slope, show that the points (−2, −1), (4, 0), (3, 3) and (−3, 2) are the vertices of a parallelogram.
Find the equation of a straight line with slope − 1/3 and y-intercept − 4.
Find the equation of a straight line with slope −2 and intersecting the x-axis at a distance of 3 units to the left of origin.
Find the equation of a line which is perpendicular to the line joining (4, 2) and (3, 5) and cuts off an intercept of length 3 on y-axis.
Find the equations of the straight lines which cut off an intercept 5 from the y-axis and are equally inclined to the axes.
Find the angles between the following pair of straight lines:
3x + y + 12 = 0 and x + 2y − 1 = 0
Find the angles between the following pair of straight lines:
3x − y + 5 = 0 and x − 3y + 1 = 0
Prove that the points (2, −1), (0, 2), (2, 3) and (4, 0) are the coordinates of the vertices of a parallelogram and find the angle between its diagonals.
Find the angle between the line joining the points (2, 0), (0, 3) and the line x + y = 1.
If θ is the angle which the straight line joining the points (x1, y1) and (x2, y2) subtends at the origin, prove that \[\tan \theta = \frac{x_2 y_1 - x_1 y_2}{x_1 x_2 + y_1 y_2}\text { and } \cos \theta = \frac{x_1 x_2 + y_1 y_2}{\sqrt{{x_1}^2 + {y_1}^2}\sqrt{{x_2}^2 + {y_2}^2}}\].
Show that the tangent of an angle between the lines \[\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\text { is } \frac{2ab}{a^2 - b^2}\].
Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.
If m1 and m2 are slopes of lines represented by 6x2 - 5xy + y2 = 0, then (m1)3 + (m2)3 = ?
Find the equation of the straight line passing through (1, 2) and perpendicular to the line x + y + 7 = 0.
If the line joining two points A(2, 0) and B(3, 1) is rotated about A in anticlock wise direction through an angle of 15°. Find the equation of the line in new position.
If one diagonal of a square is along the line 8x – 15y = 0 and one of its vertex is at (1, 2), then find the equation of sides of the square passing through this vertex.
The two lines ax + by = c and a′x + b′y = c′ are perpendicular if ______.
The equation of the line passing through (1, 2) and perpendicular to x + y + 7 = 0 is ______.
Show that the tangent of an angle between the lines `x/a + y/b` = 1 and `x/a - y/b` = 1 is `(2ab)/(a^2 - b^2)`
If the equation of the base of an equilateral triangle is x + y = 2 and the vertex is (2, – 1), then find the length of the side of the triangle.
P1, P2 are points on either of the two lines `- sqrt(3) |x|` = 2 at a distance of 5 units from their point of intersection. Find the coordinates of the foot of perpendiculars drawn from P1, P2 on the bisector of the angle between the given lines.
The equation of the straight line passing through the point (3, 2) and perpendicular to the line y = x is ______.
Equation of the line passing through (1, 2) and parallel to the line y = 3x – 1 is ______.
If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral.
The equation of the line through the intersection of the lines 2x – 3y = 0 and 4x – 5y = 2 and
| Column C1 | Column C2 |
| (a) Through the point (2, 1) is | (i) 2x – y = 4 |
| (b) Perpendicular to the line (ii) x + y – 5 = 0 x + 2y + 1 = 0 is |
(ii) x + y – 5 = 0 |
| (c) Parallel to the line (iii) x – y –1 = 0 3x – 4y + 5 = 0 is |
(iii) x – y –1 = 0 |
| (d) Equally inclined to the axes is | (iv) 3x – 4y – 1 = 0 |
The three straight lines ax + by = c, bx + cy = a and cx + ay = b are collinear, if ______.
If the line joining two points A (2, 0) and B (3, 1) is rotated about A in anticlockwise direction through an angle of 15°, then the equation of the line in new position is ______.
