Coordinate Geometry - Study Mode

[#41] A(7, -8) and C(1, 4) are vertices of a square ABCD. Find equation of diagonal BD?
Correct Answer

(B) x - 2y = 8

Explanation

Solution: By the midpoint formula, $$left( {frac{{{x_1} + {x_2}}}{2},,frac{{{y_1} + {y_2}}}{2}}
ight)$$ Midpoint of line AC $$ = left( {frac{{7 + 1}}{2},,frac{{ - 8 + 4}}{2}}
ight) = left( {4,, - 2}
ight)$$ O(x 3 , y 3 ) = (4, -2) (M 1 ) Slope of line AC, $${M_1} = frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = frac{{4 - left( { - 8}
ight)}}{{1 - 7}} = - 2$$ If the lines are ⊥, then M 1 × M 2 = -1 - 2 × M 2 = -1 M 2 = $$frac{1}{2}$$ ∴ Equation of line BD y - y 3 = M 2 (x - x 3 ) y - (-2) = $$frac{1}{2}$$(x - 4) y + 2 = $$frac{1}{2}$$(x - 4) x - 2y = 8

[#42] An equation whose graph passes through the origin, out of the given equation 2x - 3y = 3, 2x + 3y = 2, -2x + 3y = 5 and 2x + 3y = 0 is:
Correct Answer

(C) 2x + 3y = 0

Explanation

Solution: The equation whose graph passes through origin must satisfy point (0, 0) in the equation means if we put value of x & y equal to zero, the equation must be equal to zero. ∴ 2x + 3y = 0

[#43] The point P(a, b) is first reflected in origin to P 1 and P 1 is reflected in Y-axis to (4, -3). What are the co-ordinates of point P?
Correct Answer

(A) (4, 3)

Explanation

Solution: If P 2 (4, -3) is reflected through y-axis then if becomes P 1 (-4, -3) and if it is reflected through origin then it becomes P(4, 3).

[#44] What are the co-ordinates of the centroid of a triangle, whose vertices are A(2, 5), B(-4, 0) and C(5, 4)?
Correct Answer

(B) (1, 3)

Explanation

Solution: $$eqalign{
& { ext{Coordinates of centroid of a }}Delta cr
& Rightarrow left( {frac{{{x_1} + {x_2} + {x_3}}}{3},,frac{{{y_1} + {y_2} + {y_3}}}{3}}
ight) cr
& Rightarrow left( {frac{{2 + left( { - 4}
ight) + 5}}{3},,frac{{5 + 0 + 4}}{3}}
ight) cr
& Rightarrow left( {1,,3}
ight) cr} $$

[#45] The graphs of the equations $$4x + frac{1}{3}y = frac{8}{3}$$ xa0 and $$frac{1}{2}x + frac{3}{4}y + frac{5}{2} = 0$$ xa0 xa0intersect at a point P. The point P also lies on the graph of the equation:
Correct Answer

(D) 3x - y - 7 = 0

Explanation

Solution: $$eqalign{
& 4x + frac{1}{3}y = frac{8}{3} cr
& 12x + y = 8{ ext{ }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}left( { ext{i}}
ight) cr
& frac{1}{2}x + frac{3}{4}y + frac{5}{2} = 0 cr
& 2x + 3y = - 10{ ext{ }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}{ ext{. }}left( {{ ext{ii}}}
ight) cr
& { ext{Solve equation }}left( { ext{i}}
ight){ ext{ and }}left( {{ ext{ii}}}
ight) cr
& 12x + y = 8 cr
& underline {2x + 3y = - 10} ,,,,, * 6 cr
& 12x + y = 8 cr
& 12x + 18y = - 60 cr
& underline { - ,,,,, - ,,,,,,,,,,,,,,,, + ,,,} cr
& - 17y = 68 cr
& y = - 4 cr
& x = 1 cr
& P = left( {1,, - 4}
ight) cr
& { ext{Only option }}left( { ext{D}}
ight){ ext{satisfy in this point }}P cr
& 3x - y - 7 = 0 cr
& 3 imes 1 - left( { - 4}
ight) - 7 = 0 cr
& 0 = 0,,,left[ {{ ext{satisfy}}}
ight] cr} $$