Draw a quadrilateral in the Cartesian plane, whose vertices are \((-4,5), (0,7), (5,-5), (-4,-2)\). Also, find its area.
\(\dfrac{121}{2}\) square unit.
The base of an equilateral triangle with side \(2a\) lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.
(0, a), (0, -a) and \((-\sqrt{3} a, 0)\) or (0, a), (0, -a), and \((\sqrt{3} a, 0)\)
Find the distance between P \((x_1, y_1)\) and Q \((x_2, y_2)\) when (i) PQ is parallel to the y-axis, (ii) PQ is parallel to the x-axis.
(i) \(|y_2 - y_1|\), (ii) \(|x_2 - x_1|\)
Find a point on the x-axis which is equidistant from the points (7, 6) and (3, 4).
\(\left(\dfrac{15}{2}, 0\right)\)
Find the slope of a line which passes through the origin, and the mid-point of the line segment joining the points P (0, −4) and B (8, 0).
\(-\dfrac{1}{2}\)
Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (−1, −1) are the vertices of a right-angled triangle.
Find the slope of the line which makes an angle of \(30^\circ\) with the positive direction of y-axis measured anticlockwise.
\(-\sqrt{3}\)
Without using distance formula, show that points (−2, −1), (4, 0), (3, 3) and (−3, 2) are the vertices of a parallelogram.
Find the angle between the x-axis and the line joining the points (3, −1) and (4, −2).
135°
The slope of a line is double the slope of another line. If tangent of the angle between them is \(\dfrac{1}{3}\), find the slopes of the lines.
1 and 2, or \(\dfrac{1}{2}\) and 1, or −1 and −2, or −\(\dfrac{1}{2}\) and −1
A line passes through \((x_1, y_1)\) and \((h, k)\). If slope of the line is m, show that \(k - y_1 = m(h - x_1)\).
Write the equations for the x-axis and y-axis.
For the x-axis: \( y = 0 \); for the y-axis: \( x = 0 \).
Find the equation of the line passing through the point \( (-4, 3) \) with slope \( \dfrac{1}{2} \).
\( x - 2y + 10 = 0 \)
Find the equation of the line passing through \( (0, 0) \) with slope \( m \).
\( y = mx \)
Find the equation of the line passing through \( (2, 2\sqrt{3}) \) and inclined with the x-axis at an angle of \( 75^\circ \).
\( (\sqrt{3} + 1)x - (\sqrt{3} - 1)y = 4(\sqrt{3} - 1) \)
Find the equation of the line intersecting the x-axis at a distance of 3 units to the left of the origin with slope \( -2 \).
\( 2x + y + 6 = 0 \)
Find the equation of the line intersecting the y-axis at a distance of 2 units above the origin and making an angle of \( 30^\circ \) with the positive direction of the x-axis.
\( x - \sqrt{3}y + 2\sqrt{3} = 0 \)
Find the equation of the line passing through the points \( (-1, 1) \) and \( (2, -4) \).
\( 5x + 3y + 2 = 0 \)
The vertices of \( \triangle PQR \) are \( P(2, 1) \), \( Q(-2, 3) \) and \( R(4, 5) \). Find the equation of the median through the vertex \( R \).
\( 3x - 4y + 8 = 0 \)
Find the equation of the line passing through \( (-3, 5) \) and perpendicular to the line through the points \( (2, 5) \) and \( (-3, 6) \).
\( 5x - y + 20 = 0 \)
A line perpendicular to the line segment joining the points \( (1, 0) \) and \( (2, 3) \) divides it in the ratio \( 1 : n \). Find the equation of this line.
\( (1 + n)x + 3(1 + n)y = n + 11 \)
Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point \( (2, 3) \).
\( x + y = 5 \)
Find the equations of the lines passing through the point \( (2, 2) \) and cutting off intercepts on the axes whose sum is 9.
\( x + 2y - 6 = 0 \) and \( 2x + y - 6 = 0 \)
Find the equation of the line through the point \( (0, 2) \) making an angle \( \dfrac{2\pi}{3} \) with the positive x-axis. Also, find the equation of the line parallel to it and crossing the y-axis at a distance of 2 units below the origin.
First line: \( \sqrt{3}x + y - 2 = 0 \); parallel line through \( (0, -2) \): \( \sqrt{3}x + y + 2 = 0 \)
The perpendicular from the origin to a line meets it at the point \( (-2, 9) \). Find the equation of the line.
\( 2x - 9y + 85 = 0 \)
The length \( L \) (in centimetre) of a copper rod is a linear function of its Celsius temperature \( C \). In an experiment, \( L = 124.942 \) when \( C = 20 \) and \( L = 125.134 \) when \( C = 110 \). Express \( L \) in terms of \( C \).
\( L = \dfrac{192}{90}(C - 20) + 124.942 \)
The owner of a milk store finds that he can sell 980 litres of milk each week at Rs 14 per litre and 1220 litres each week at Rs 16 per litre. Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at Rs 17 per litre?
\( 1340 \) litres
\( P(a, b) \) is the mid-point of a line segment between the axes. Show that the equation of the line is \( \dfrac{x}{a} + \dfrac{y}{b} = 2 \).
Point \( R(h, k) \) divides a line segment between the axes in the ratio \( 1 : 2 \). Find the equation of the line.
\( 2kx + hy = 3kh \)
By using the concept of the equation of a line, prove that the three points \( (3, 0) \), \( (-2, -2) \) and \( (8, 2) \) are collinear.
Reduce the following equations into slope-intercept form and find their slopes and the \(y\)-intercepts:
(i) \(x + 7y = 0\)
(ii) \(6x + 3y - 5 = 0\)
(iii) \(y = 0\)
(i) \(y = -\dfrac{1}{7}x + 0,\; -\dfrac{1}{7},\; 0\)
(ii) \(y = -2x + \dfrac{5}{3},\; -2,\; \dfrac{5}{3}\)
(iii) \(y = 0x + 0,\; 0,\; 0\)
Reduce the following equations into intercept form and find their intercepts on the axes:
(i) \(3x + 2y - 12 = 0\)
(ii) \(4x - 3y = 6\)
(iii) \(3y + 2 = 0\)
(i) \(\dfrac{x}{4} + \dfrac{y}{6} = 1,\; 4,\; 6\)
(ii) \(\dfrac{x}{3} + \dfrac{y}{-2} = 1,\; \dfrac{3}{2},\; -2\)
(iii) \(y = -\dfrac{2}{3},\; \) intercept with \(y\)-axis \(= -\dfrac{2}{3}\) and no intercept with \(x\)-axis
Find the distance of the point \((-1, 1)\) from the line \(12(x + 6) = 5(y - 2)\).
5 units
Find the points on the \(x\)-axis whose distances from the line \(\dfrac{x}{3} + \dfrac{y}{4} = 1\) are 4 units.
\((-2, 0)\) and \((8, 0)\)
Find the distance between parallel lines:
(i) \(15x + 8y - 34 = 0\) and \(15x + 8y + 31 = 0\)
(ii) \(l(x + y) + p = 0\) and \(l(x + y) - r = 0\)
(i) \(\dfrac{65}{17}\) units
(ii) \(\dfrac{1}{\sqrt{2}} \left| \dfrac{p + r}{l} \right|\) units
Find equation of the line parallel to the line \(3x - 4y + 2 = 0\) and passing through the point \((-2, 3)\).
\(3x - 4y + 18 = 0\)
Find equation of the line perpendicular to the line \(x - 7y + 5 = 0\) and having \(x\)-intercept 3.
\(y + 7x = 21\)
Find angles between the lines \(\sqrt{3}x + y = 1\) and \(x + \sqrt{3}y = 1\).
\(30^\circ\) and \(150^\circ\)
The line through the points \((h, 3)\) and \((4, 1)\) intersects the line \(7x - 9y - 19 = 0\) at right angle. Find the value of \(h\).
\(\dfrac{22}{9}\)
Prove that the line through the point \((x_1, y_1)\) and parallel to the line \(Ax + By + C = 0\) is \(A(x - x_1) + B(y - y_1) = 0\).
Two lines passing through the point \((2, 3)\) intersect each other at an angle of \(60^\circ\). If slope of one line is 2, find equation of the other line.
\((\sqrt{3} + 2)x + (2\sqrt{3} - 1)y = 8\sqrt{3} + 1\)
or
\((\sqrt{3} - 2)x + (1 + 2\sqrt{3})y = -1 + 8\sqrt{3}\)
Find the equation of the right bisector of the line segment joining the points \((3, 4)\) and \((-1, 2)\).
\(2x + y = 5\)
Find the coordinates of the foot of perpendicular from the point \((-1, 3)\) to the line \(3x - 4y - 16 = 0\).
\(\left(\dfrac{68}{25}, -\dfrac{49}{25}\right)\)
The perpendicular from the origin to the line \(y = mx + c\) meets it at the point \((-1, 2)\). Find the values of \(m\) and \(c\).
\(m = \dfrac{1}{2},\; c = \dfrac{5}{2}\)
If \(p\) and \(q\) are the lengths of perpendiculars from the origin to the lines \(x \cos \theta - y \sin \theta = k \cos 2\theta\) and \(x \sec \theta + y \csc \theta = k\), respectively, prove that \(p^2 + 4q^2 = k^2\).
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 \(A\).
\(y - x = 1,\; \sqrt{2}\)
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
\(\displaystyle \dfrac{1}{p^2} = \dfrac{1}{a^2} + \dfrac{1}{b^2}\).
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