X-axis will act as a plane mirror, and this point will form an image, following the sign convention, at (4, -5) in the fourth quadrant.
The points on the line should satisfy the equation of the line.
So, the point P (a, y) satisfies the equation.
Hence,
3x+ 4y – 12 = 0
3(a) + 4(y) -12 = 0
4y = 12-3a
Y = $\dfrac{(12-3a) }{ 4}$
Since it is given that the point P lies on the line y = -1, its y-coordinate will be -1, and the x-coordinate can be any real number.
Let the origin be O and the point A be (-2, -2)
Using the distance formula,
$OA^{2}$ = $(2^{2} + 2^{2})$
$OA^{2}$ = 8
OA=$\sqrt{8}$=2$\sqrt{2}$
Let A (a, b) and B (-a,-b) be the two points and ’d’ be the distance between them.
By using the distance formula, we get
d=$\sqrt{(-a(-a))^{2}+(-b(-b))^{2}}$
d=$\sqrt{(2a)^{2}+(2b)^{2}}$
d=2$\sqrt{a^{2}+b^{2}}$
Midpoint of a line segment joining (x1, y1) and (x2, y2) is [(x1+ x2)/2, (y1+y2)/2]
∴ Mid-point of the line-segment joining the points (–5, 4) and (9, –8) = [(9-5)/2, (4-8)/2] = (2, -2)
The coordinates for the point that divides a line in the ratio m:n is
$\dfrac{n\times x_{1}+m\times x_{2}}{m+n}$ , $\dfrac{n\times y_{1}+m\times y_{2}}{n+m}$
Substituting in the equation, we get
=$\dfrac{3\times 3+4\times 1}{3+4}$ , $\dfrac{3\times 4+4\times 2}{3+4}$
= 13/7, 20/7
Therefore, the point which divides the line segment in the ratio 3:4 is 13/7, 20/7.
The given points are: A(1,2), B(4, y), C(x, 6), and D(3,5)
Since the diagonals of a parallelogram bisect each other,
The coordinates of P are:
X= $\dfrac{(x+1)}{ 2}$ = $\dfrac{(3+4)}{2}$
⇒x+1=7⇒x=6
Y = (5+Y)/2= (6+2)/2
⇒5+y=8⇒y=3
∴ The required values of x and y are:
x=6, y=3.
Area of a triangle formed by (x1, y1), (x2, y2), (x3, y3) = 1/2 [x1(y2 – y3) + x2(y3 – y1)+ x3(y1 – y2)]
For 3 points to be collinear, the area of the triangle should be zero:
⇒1/2[a (b−1) + 0(1−0) + 1(0−b)] =0
⇒1/2[a (b−1) +1(0−b)] =0
⇒ab=a+b
⇒ (1/a) + (1/b) = 1
$\dfrac{Area\:\: of\:\: triangle 1 }{Area \:\: of\:\: triangle 2}$
=$\dfrac{ (\dfrac{1}{2} × Base 1× height) }{ (\dfrac{1}{2} × Base 2× height) }$
= $\dfrac{ base 1}{base 2}$