Question

The abscissa of a point A is equal to its ordinate,and its distance fro the point B(1,3) is 10 units, what are the co ordinates of A

Answer 1

we know that x-coordinate is also called as abscissa and y-coordinate is also called as ordinate, so abscissa = ordinate then x-coordinate = y-coordinate

let the point be (x,x) and distance between (x,x) and (1,3) is 10

then by applying distance formula:

(x-1)^2 + (x-3)^2 = 100

solving we get a quadratic equation x^2 - 4x - 45 = 0

solving we get x = -5 and x = 9

so the point A can be (-5 , -5) or (9 ,9)

Answer 2

let the coordinates be A(x,x)
AB= 10
$\sqrt{ {(x1 - x2)}^{2} + {(y1 - y2)}^{2} } = 10 \\ \sqrt{ {(x - 1)}^{2} + {(x - 3)}^{2} } = 10 \\ {(x - 1)}^{2} + {(x - 3)}^{2} = 100 \\ 2 {x}^{2} - 8x + 10 = 100 \\ {x}^{2} - 4x + 5 = 50 \\ {x}^{2} - 4x - 45 = 0 \\ {x}^{2} - 9x + 5x - 45 = 0 \\ x(x - 9) + 5(x - 9) = 0 \\ (x + 5)(x - 9) = 0 \\ x = - 5 \\ x = 9$
therefore coordinates are A(-5,-5)or A(9,9)