NCERT Solutions

Draw a quadrilateral in the Cartesian plane, whose vertices are $(-4, 5), (0, 7), (5, -5)$ and $(-4, -2)$. Also, find its area.

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.

Find the distance between $\mathrm{P}(x_{1}, y_{1})$ and $\mathrm{Q}(x_{2}, y_{2})$ when : (i) PQ is parallel to the $y$-axis, (ii) PQ is parallel to the $x$-axis.

Find a point on the $x$-axis, which is equidistant from the points $(7, 6)$ and $(3, 4)$.

Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points $\mathrm{P}(0,-4)$ and $\mathrm{B}(8,0)$.

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.

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 $(4, -2)$.

The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$, find the slopes of the lines.

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 $x$-and $y$-axes.

Passing through the point $(-4, 3)$ with slope $\frac{1}{2}$.

Passing through $(0, 0)$ with slope $m$.

Passing through $(2, 2\sqrt{3})$ and inclined with the $x$-axis at an angle of $75^{\circ}$.

Intersecting the $x$-axis at a distance of 3 units to the left of origin with slope -2.

Intersecting the $y$-axis at a distance of 2 units above the origin and making an angle of $30^{\circ}$ with positive direction of the $x$-axis.

Passing through the points $(-1, 1)$ and $(2, -4)$.

The vertices of $\triangle \mathrm{PQR}$ are $\mathrm{P}(2, 1), \mathrm{Q}(-2, 3)$ and $\mathrm{R}(4, 5)$. Find equation of the median through the vertex R.

Find the equation of the line passing through $(-3, 5)$ and perpendicular to the line through the points $(2, 5)$ and $(-3, 6)$.

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 the line.

Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point $(2, 3)$.

Find equation of the line passing through the point $(2, 2)$ and cutting off intercepts on the axes whose sum is 9.

Find equation of the line through the point $(0, 2)$ making an angle $\frac{2 \pi}{3}$ with the positive $x$-axis. Also, find the equation of line parallel to it and crossing the $y$-axis at a distance of 2 units below the origin.

The perpendicular from the origin to a line meets it at the point $(-2, 9)$, find the equation of the line.

The length L (in centimetre) of a copper rod is a linear function of its Celsius temperature C. In an experiment, if $\mathrm{L}=124.942$ when $\mathrm{C}=20$ and $\mathrm{L}=125.134$ when $\mathrm{C}=110$, express L in terms of C.

The owner of a milk store finds that, he can sell 980 litres of milk each week at Rs 14/litre and 1220 litres of milk each week at Rs 16/litre. Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at Rs 17/litre?

$\mathrm{P}(a, b)$ is the mid-point of a line segment between axes. Show that equation of the line is $\frac{x}{a}+\frac{y}{b}=2$.

Point $\mathrm{R}(h, k)$ divides a line segment between the axes in the ratio $1:2$. Find equation of the line.

By using the concept of equation of a line, prove that the three points $(3, 0), (-2, -2)$ and $(8, 2)$ are collinear.

Find the distance of the point $(-1, 1)$ from the line $12(x + 6) = 5(y - 2)$.

Find the points on the $x$-axis, whose distances from the line $\frac{x}{3} + \frac{y}{4} = 1$ are 4 units.

Find equation of the line parallel to the line $3x - 4y + 2 = 0$ and passing through the point $(-2, 3)$.

Find equation of the line perpendicular to the line $x - 7y + 5 = 0$ and having $x$ intercept 3.

Find angles between the lines $\sqrt{3}x + y = 1$ and $x + \sqrt{3}y = 1$.

The line through the points $(h, 3)$ and $(4, 1)$ intersects the line $7x - 9y - 19 = 0$. at right angle. Find the value of $h$.

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 $(2, 3)$ intersects each other at an angle of $60^{\circ}$. If slope of one line is 2, find equation of the other line.

Find the equation of the right bisector of the line segment joining the points $(3, 4)$ and $(-1, 2)$.

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 $p$ and $q$ are the lengths of perpendiculars from the origin to the lines $x \cos \theta - y \sin \theta = k \cos 2\theta$ and $x \sec \theta + y \operatorname{cosec} \theta = k$, respectively, prove that $p^2 + 4q^2 = k^2$.

In the triangle ABC with vertices $\mathrm{A}(2, 3), \mathrm{B}(4, -1)$ and $\mathrm{C}(1, 2)$, find the equation and length of altitude from the vertex A.

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 $\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}$.

Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and -6, respectively.

What are the points on the $y$-axis whose distance from the line $\frac{x}{3} + \frac{y}{4} = 1$ is 4 units.

Find perpendicular distance from the origin to the line joining the points $(\cos \theta, \sin \theta)$ and $(\cos \phi, \sin \phi)$.

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 $3x + y = 0$.

Find the equation of a line drawn perpendicular to the line $\frac{x}{4} + \frac{y}{6} = 1$ through the point, where it meets the $y$-axis.

Find the area of the triangle formed by the lines $y - x = 0, x + y = 0$ and $x - k = 0$.

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.

If three lines whose equations are $y = m_1 x + c_1, y = m_2 x + c_2$ and $y = m_3 x + c_3$ are concurrent, then show that $m_1(c_2 - c_3) + m_2(c_3 - c_1) + m_3(c_1 - c_2) = 0$.

Find the equation of the lines through the point $(3, 2)$ which make an angle of $45^{\circ}$ with the line $x - 2y = 3$.

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.

Show that the equation of the line passing through the origin and making an angle $\theta$ with the line $y = mx + c$ is $\frac{y}{x} = \frac{m \pm \tan \theta}{1 \mp m \tan \theta}$.

In what ratio, the line joining $(-1, 1)$ and $(5, 7)$ is divided by the line $x + y = 4$?

Find the distance of the line $4x + 7y + 5 = 0$ from the point $(1, 2)$ along the line $2x - y = 0$.

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 $x + y = 4$ may be at a distance of 3 units from this point.

The hypotenuse of a right angled triangle has its ends at the points $(1, 3)$ and $(-4, 1)$. Find an equation of the legs (perpendicular sides) of the triangle which are parallel to the axes.

Find the image of the point $(3, 8)$ with respect to the line $x + 3y = 7$ assuming the line to be a plane mirror.

If the lines $y = 3x + 1$ and $2y = x + 3$ are equally inclined to the line $y = mx + 4$, find the value of $m$.

If sum of the perpendicular distances of a variable point $\mathrm{P}(x, y)$ from the lines $x + y - 5 = 0$ and $3x - 2y + 7 = 0$ is always 10. Show that P must move on a line.

Find equation of the line which is equidistant from parallel lines $9x + 6y - 7 = 0$ and $3x + 2y + 6 = 0$.

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.

Prove that the product of the lengths of the perpendiculars drawn from the points $(\sqrt{a^2 - b^2}, 0)$ and $(-\sqrt{a^2 - b^2}, 0)$ to the line $\frac{x}{a} \cos \theta + \frac{y}{b} \sin \theta = 1$ is $b^2$.

A person standing at the junction (crossing) of two straight paths represented by the equations $2x - 3y + 4 = 0$ and $3x + 4y - 5 = 0$ wants to reach the path whose equation is $6x - 7y + 8 = 0$ in the least time. Find equation of the path that he should follow.