A ray of light coming from the point (1, 2)is reflected at a point A on the x-axis and it passes through the point (5, 3). Find the co-ordinates of the point A.

A line is drawn perpendicular to 5x = y + 7. Find the equation of the line if the area of the triangle formed by this line with co-ordinate axes is 10 sq. units.

Show that the points (1, 3), (2, 1) and \((\frac{1}{2},4)\) are collinear, by using
(i) concept of slope
(ii) a straight line
(iii) any other method.

In a shopping mall there is a hall of cuboid shape with dimension 800 \(\times\)800 \(\times\)720 units, which needs to be added the facility of an escalator in the path as shown by the dotted line in the figure. Find 
(i) the minimum total length of the escalator.
(ii) the heights at which the escalator changes its direction.
(iii) the slopes of the escalator at the turning points.

Find the equation of the perpendicular bisector of the line segment joining the points (1, 1) and (2, 3).

Suppose the Government has decided to erect a new Electrical Power Transmission Substation to provide better power supply to two villages namely A and B. The substation has to be on the line l. The distances of villages A and B from the foot of the perpendiculars P and Q on the line l are 3 km and 5 km respectively and the distance between P and Q is 6 km. (i) What is the smallest length of cable required to connect the two villages.
(ii) Find the equations of the cable lines that connect the power station to two villages. (Using the knowledge in conjunction with the principle of reflection allows for approach to solve this problem.)

A car rental firm has charges Rs. 25 with 1.8 free kilometers, and Rs. 12 for every additional kilometer. Find the equation relating the cost y to the number of kilometers x. Also find the cost to travel 15 kilometers.

Consider a hollow cylindrical vessel, with circumference 24 cm and height 10 cm. An ant is located on the outside of vessel 4 cm from the bottom. There is a drop of honey at the diagrammatically opposite inside of the vessel, 3 cm from the top. 
(i) What is the shortest distance the ant would need to crawl to get the honey drop?
(ii) Equation of the path traced out by the ant.
(iii) Where the ant enter in to the cylinder? Here is a picture that illustrates the position of the ant and the honey.

Express the equation √3x - y + 4 = 0 in the following equivalent form Intercept form,

Find the distance
between two parallel lines 3x + 4y = 12 and 6x + 8y + 1 = 0.