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Q : 20    By using the concept of equation of a line, prove that the three points  $(3,0),(-2,-2)$ and  $(8,2)$  are collinear.

Points are collinear means they lies on same line Now,  given points are    and   Equation of line passing through point A and B is Therefore, the equation of line passing through A and B is  Now, Equation of line passing through point B and C is Therefore, Equation of line passing through point B and C is  When can clearly see that  Equation of line passing through point A nd B ...

Q : 19     Point  $R(h,k)$  divides a line segment between the axes in the ratio $1:2$ .  Find  equation of the line.

Let the coordinates of Point A is (x,0) and of point B is (0,y) It is given that point R(h , k) divides the line segment between the axes in the ratio  Therefore, R(h , k)  Therefore, coordinates of point A is   and of point B is  Now, slope of line passing through points  and   is  Now, equation of line passing through point   and with slope  is  Therefore, the equation of line is

Q: 18        $P(a,b)$  is the mid-point of a line segment between axes.  Show that equation
of the line is  $\frac{x}{a}+\frac{y}{b}=2$.

Now, let coordinates of point A is (0 , y) and of point B is (x , 0) The, Therefore, the coordinates of point A is (0 , 2b) and of point B is (2a , 0) Now, slope of line passing through points (0,2b) and (2a,0) is Now, equation of line passing through point (2a,0) and with slope    is Hence proved

Q : 17     The owner of a milk store finds that, he can sell 980 litres of milk each week at $Rs\hspace{1mm}14/litre$  and  $1220$  litres of milk each week at  $Rs\hspace{1mm}16/litre$ .  Assuming a linear relationship between selling price and demand, how many litres could he sell weekly at  $Rs\hspace{1mm}17/litre$ ?

It is given that the owner of a milk store sell 980 litres milk each week at  and    litres of milk each week at   Now, if we assume the rate of milk as x-axis and Litres of milk as y-axis Then, we will get coordinates of two points i.e.  (14, 980)  and   (16, 1220) Now, the relation between  litres of milk and Rs/litres is given by equation  Now, at  he could sell He could sell...

Q : 16         The length  $L$(in centimetre) of a copper rod is a linear function of its Celsius  temperature $C$.  In an experiment, if  $L=124.942$  when $C=20$ and  $L=125.134$ when $C=110$, express $L$ in terms of $C$

It is given that If  then  and  If    then   Now, if assume C along x-axis and L along y-axis Then, we will get coordinates of two points (20 , 124.942)  and (110 , 125.134) Now, the relation between C and L is given by equation Which is the required relation

Q: 15     The perpendicular from the origin to a line meets it at the point $(-2,9)$ , find the equation of the line.

Let the slope of the line is m and slope of a perpendicular line is which passes through the origin (0, 0) and (-2, 9) is Now, the slope of the line is Now, the equation of line passes through the point (-2, 9) and with slope   is Therefore, the equation of the line is

Q : 14     . 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.

We know that  Now, equation of line passing through point (0 , 2) and with slope  is Therefore, equation of line is                   -(i) Now, It is given that line crossing the -axis at a distance of units below the origin which means coordinates are  (0 ,-2) This line is parallel to above line which means slope of both the lines are equal Now, equation of line passing through point...

Q : 13         Find equation of the line passing through the point  $(2,2)$ and cutting off intercepts on the axes whose sum is $9$.

Let (a, b) are the intercept on x and y axis respectively Then, the equation of line is given by It is given that a + b = 9  b = 9 - a Now, It is given that line passes through point (2 ,2) So, case (i)  a = 6  b = 3   case (ii)   a = 3 , b = 6 Therefore, equation of line is 2x + y = 6 , x + 2y = 6

Q: 12         Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point  $(2,3)$.

Let (a, b) are the intercept on x and y-axis respectively Then, the equation of the line is given by Intercepts are equal which means a = b Now, it is given that line passes through the point (2,3) Therefore, therefore, equation of the line is

Q : 11          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.

Co-ordinates of point which divide line segment joining the points    and    in the ratio  is Let the slope of the perpendicular line is m And Slope of  line segment joining the points    and   is Now, slope of perpendicular line is Now, equation of line passing through point  and with slope m is equation of line passing through point  and with slope  is Therefore, equation of line  is

Q : 10         Find the equation of the line passing through  $(-3,5)$  and perpendicular to the line through the points  $(2,5)$  and $(-3,6)$.

It is given that the line passing through    and perpendicular to the line through the points    and Let the slope of the line passing through the point (-3,5) is m and Slope of line  passing through points (2,5) and (-3,6) Now this line is perpendicular to line passing through point (-3,5) Therefore, Now, equation of line passing through point  and with slope m is equation of line...

Q : 9         The vertices of  $\Delta \hspace{1mm}PQR$ are  $P(2,1),Q(-2,3)$  and   $R(4,5)$. Find equation of the median through the vertex $R$

The vertices of   are    and    Let m be RM b the median through vertex R Coordinates of M (x, y ) =  Now, slope of line RM Now, equation of line passing through point  and with slope m is equation of line passing through point (0 , 2) and with slope  is Therefore, equation of median is

Find the equation of the line which satisfy the given conditions:

Q : 8         Perpendicular distance from the origin is $5$ units and the angle made by the  perpendicular with the positive $x$-axis is  $30^{\circ}$

It is given that length of perpendicular is 5 units  and  angle made by the  perpendicular with the positive -axis is   Therefore, equation of line is In this case p = 5 and   Therefore, equation of the line  is

Find the equation of the line which satisfy the given conditions:

Q : 7         Passing through the points  $(-1,1)$  and $(2,-4)$.

We know that , equation of line passing through point  and with slope m is given by Now, it is given that line passes throught point (-1 ,1) and (2 , -4) Now,  equation of line passing through point (-1,1) and with slope   is

Find the equation of the line which satisfy the given conditions:

Q : 6         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.

We know that , equation of line passing through point  and with slope m is given by Line Intersecting the y-axis at a distance of 2 units above the origin which means point is (0,2) we know that Now, the equation of the line passing through the point (0,2) and with slope   is  Therefore, the equation of the line  is

Find the equation of the line which satisfy the given conditions:

Q : 5         Intersecting the $x$-axis at a distance of $3$ units to the left of origin with slope $-2$

We know that the equation of the line passing through the point  and with slope m is given by Line Intersecting the -axis at a distance of  units to the left of origin which means the point is (-3,0) Now, the equation of the line passing through the point (-3,0) and with slope -2  is  Therefore, the equation of the line  is

Find the equation of the line which satisfy the given conditions:

Q : 4         Passing through  $(2,2\sqrt{3})$  and inclined with the x-axis at an angle of $75^{\circ}$.

We know that the equation of the line passing through the point  and with slope m is given by we know that where  is angle made by line with positive x-axis measure in the anti-clockwise direction Now, the equation of the line passing through the point  and with slope   is  Therefore, the equation of the line  is

Find the equation of the line which satisfy the given conditions:

Q : 3    Passing through $(0,0)$ with slope $m$.

We know that the equation of the line passing through the point  and with slope m is given by Now, the equation of the line passing through the point (0,0) and with slope m is  Therefore, the equation of the line  is

Find the equation of the line which satisfy the given conditions:

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

We know that , equation of line passing through point  and with slope m is given by Now,  equation of line passing through point (-4,3) and with slope  is  Therefore, equation of the line  is

Find the equation of the line which satisfy the given conditions:

Q : 1     Write the equations for the  $x$-and  $y$-axes.

Equation of x-axis is y = 0 and  Equation of y-axis is x = 0
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