Coordinate Geometry - Study Mode

[#11] The distance between the points (4, 8) and (k, -4) is 13. What is the value of k?
Correct Answer

(C) -1

Explanation

Solution: Distance formula between two points Distance = $$sqrt {{{left( {{x_2} - {x_1}}
ight)}^2} + {{left( {{y_2} - {y_1}}
ight)}^2}} $$ ⇒ 13 = $$sqrt {{{left( {k - 4}
ight)}^2} + {{left( { - 4 - 8}
ight)}^2}} $$ ⇒ 169 = k 2 + 16 - 8k + 144 ⇒ k 2 - 8k - 9 = 0 ⇒ k 2 - 9k + k - 9 = 0 ⇒ k(k - 9) + 1(k - 9) = 0 ⇒ (k + 1)(k - 9) = 0 ⇒ k = -1 and 9 ∴ k = -1 (According to options)

[#12] The slope of the line passing through the points (2, -1) and (x, 5) is -1. Find x?
Correct Answer

(B) -4

Explanation

Solution: $$eqalign{
& { ext{Slope}} Rightarrow m = - 1 cr
& m = frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} cr
& - 1 = frac{{5 - left( { - 1}
ight)}}{{x - 2}} cr
& - 1left( {x - 2}
ight) = 6 cr
& - x + 2 = 6 cr
& x = 2 - 6 cr
& x = - 4 cr} $$

[#13] For what value of m will the system of equations 17x + my + 102 = 0 and 23x + 299y + 138 = 0 have infinite number of solutions?
Correct Answer

(C) 221

Explanation

Solution: $$eqalign{
& 17x + my + 102 = 0 cr
& 23x + 299y + 138 = 0 cr
& { ext{Infinite solution}} cr
& frac{{{a_1}}}{{{a_2}}} = frac{{{b_1}}}{{{b_2}}} cr
& frac{{17}}{{23}} = frac{m}{{299}} cr
& m = frac{{17 imes 299}}{{23}} = 221 cr} $$

[#14] For what value of k, the system of equations kx + 2y = 2 and 3x + y = 1 will be coincident?
Correct Answer

(D) 6

Explanation

Solution: $$eqalign{
& { ext{For coincident lines}} cr
& frac{{{a_1}}}{{{a_2}}} = frac{{{b_1}}}{{{b_2}}} = frac{{{c_1}}}{{{c_2}}} cr
& herefore frac{k}{3} = frac{2}{1} = frac{2}{1} cr
& { ext{Hence, }}k = 3 imes 2 cr
& k = 6 cr} $$

[#15] The area (in sq. units) of the triangle formed by the graphs of 8x + 3y = 24, 2x + 8 = y and the x-axis is:
Correct Answer

(A) 28

Explanation

Solution: $$eqalign{
& { ext{At }}x{ ext{ - axis}},,y = 0 cr
& 8x + 3y = 24 cr
& x = 3 cr
& 2x + 8 = y cr
& x = - 4 cr
& ,,,8x + 3y = 24 o left( { ext{i}}
ight) cr
& ,,,8x - 4y = 32 o left( {{ ext{ii}}}
ight) cr
& underline {, - ,,,,, + ,,,,,,,,, + ,,,,,,,,} cr
& 7y = 56 cr
& y = 8 cr
& { ext{Area}} = frac{1}{2} imes { ext{base}} imes { ext{height}} cr
& = frac{1}{2} imes 7 imes 8 cr
& = 28 cr} $$