Q12 Find an equation for the line with the given properties

Q12. Find an equation for the line with the given properties. Express the answer using the general form of the equation of a line.

Containing the points (-4, -2) and (0, -9)
   a. 7x - 4y = 36
   b. -7x - 4y = 36
   c. 2x - 9y = -81
   d. -2x + 9y = -81

Q13. Without solving, determine the character of the solutions of the equation in the complex number system.

x² + 5x + 8 = 0
   a. a repeated real solution
   b. two unequal real solutions
   c. two complex solutions that are conjugates of each other

Solution

12. The general form of the equation of a line is ax + by + c = 0 ….(1), where a, b, c are constants. Since the line passes through the point (-4, -2), on substituting x = -4 and y = -2 in the equation of the line, we get -4a -2b + c = 0…(2)

Similarly, since the line passes through the point (0, -9), on substituting x = 0 and y = -9 in the equation of the line, we get -9b + c = 0 or, b = c/9 …(3) On substituting b =c/9 in the 2^nd equation, we get -4a-2c/9 + c = 0. Therefore, – 4a = 2c/9 – c = -7c/9 so that a = 7c/36.

Now, on substituting a = c/36 and b = c/9 in the 1^st equation, we get the equation of the required line as (7c/36)x + (c/9) y + c = 0. On multiplying both the sides by 36, we get 7cx + 4cy + 36c = 0. On dividing both the sides by c, we get 7x + 4y + 36 = 0 (assuming c 0; if c = 0, we can start with the equation ax + by = 0 and the process becomes easier and shorter). Thus the required equation is –7x – 4y = 36. The answer b is correct.

13. The general form of a quadratic equation is ax² + bx + c = 0. Its roots are {–b ± ( b² – 4ac)}/2. Here a = 1, b= 5 and c = 8 so that b² – 4ac = 25 – 32 = -7. Therefore ( b² – 4ac) = (- 7) = i 7. Thus, the given equation will have complex roots and since complex roots of a quadratic equation are always conjugates, the given equation will have two complex solutions that are conjugates of each other. The answer c is correct.