Find the values of \(k\) for which the line \((k-3)x - (4-k^2)y + k^2 - 7k + 6 = 0\) is:
(a) parallel to the x-axis,
(b) parallel to the y-axis,
(c) passing through the origin.
(a) 3
(b) \(\pm 2\)
(c) 6 or 1
Find the equations of the lines which cut off intercepts on the axes whose sum and product are 1 and −6 respectively.
\(2x - 3y = 6\), \(-3x + 2y = 6\)
What are the points on the y-axis whose distance from the line \(\dfrac{x}{3} + \dfrac{y}{4} = 1\) is 4 units?
\((0, -\dfrac{8}{3}), (0, \dfrac{32}{3})\)
Find perpendicular distance from the origin to the line joining the points \((\cos\theta, \sin\theta)\) and \((\cos\phi, \sin\phi)\).
\( \left| \cos\dfrac{\phi - \theta}{2} \right| \)
Find the equation of the line parallel to y-axis and drawn through the point of intersection of the lines \(x - 7y + 5 = 0\) and \(3x + y = 0\).
\(x = -\dfrac{5}{22}\)
Find the equation of a line drawn perpendicular to the line \(\dfrac{x}{4} + \dfrac{y}{6} = 1\) through the point where it meets the y-axis.
\(2x - 3y + 18 = 0\)
Find the area of the triangle formed by the lines \(y - x = 0\), \(x + y = 0\) and \(x - k = 0\).
\(k^2\) square units
Find the value of \(p\) so that the three lines \(3x + y - 2 = 0\), \(px + 2y - 3 = 0\) and \(2x - y - 3 = 0\) may intersect at one point.
5
Find the equation of the lines through the point \((3,2)\) which make an angle of 45° with the line \(x - 2y = 3\).
3x − y = 7, x + 3y = 9
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.
13x + 13y = 6
Find the equation of the right bisector of the line segment joining the points \((3,4)\) and \((-1,2)\).
2x + y = 5
In what ratio is the line joining \((-1,1)\) and \((5,7)\) divided by the line \(x + y = 4\)?
1 : 2
Find the distance of the line \(4x + 7y + 5 = 0\) from the point \((1,2)\) along the line \(2x - y = 0\).
\(\dfrac{23\sqrt{5}}{18}\) units
Find the direction in which a straight line must be drawn through the point \((-1,2)\) so that its point of intersection with the line \(x + y = 4\) is at a distance of 3 units from this point.
The line is parallel to x-axis or parallel to y-axis.
The hypotenuse of a right-angled triangle has its ends at \((1,3)\) and \((-4,1)\). Find the equations of the legs of the triangle which are parallel to the axes.
x = 1, y = 1 or x = -4, y = 3
Find the image of the point \((3,8)\) with respect to the line \(x + 3y = 7\) assuming the line to be a plane mirror.
(−1, −4)
If the lines \(y = 3x + 1\) and \(2y = x + 3\) are equally inclined to the line \(y = mx + 4\), find the value of \(m\).
\(\dfrac{1 \pm 5\sqrt{2}}{7}\)
Find the equation of the line which is equidistant from the parallel lines \(9x + 6y - 7 = 0\) and \(3x + 2y + 6 = 0\).
18x + 12y + 11 = 0
A ray of light passing through the point \((1,2)\) reflects on the x-axis at point A and the reflected ray passes through the point \((5,3)\). Find the coordinates of A.
\(\left(\dfrac{13}{5}, 0\right)\)
A person standing at the junction of two straight paths represented by \(2x - 3y + 4 = 0\) and \(3x + 4y - 5 = 0\) wants to reach the path whose equation is \(6x - 7y + 8 = 0\) in the least time. Find the equation of the path that he should follow.
119x + 102y = 125