Question

Find the point at which the circle \[{x^2} + {y^2} - 3x - 4y + 2 = 0\] cuts the x-axis.
A. \[(2,0),( - 3,0)\]
B. \[(3,0),(4,0)\]
C. \[(1,0),( - 1,0)\]
D. \[(1,0),(2,0)\]

Answer 1

Hint: Substitute 0 for y in \[{x^2} + {y^2} - 3x - 4y + 2 = 0\] to obtain the point at which the circle cuts the x-axis. Now, solve the quadratic equation of x by the middle term factor. Express \[ - 3x\] as \[ - 2x - x\] then factor out the terms as required and obtain the values of x.

Complete step by step solution:
Substitute 0 for y in the given equation of circle \[{x^2} + {y^2} - 3x - 4y + 2 = 0\] to obtain the quadratic equation of x because at x axis value of y coordinate will be 0.
So,
\[{x^2} + 0 - 3x - 4.0 + 2 = 0\]
\[ \Rightarrow {x^2} - 3x + 2 = 0\]
Express \[ - 3x\] as \[ - 2x - x\] in the equation \[{x^2} - 3x + 2 = 0\] for further calculation.
\[{x^2} - 2x - x + 2 = 0\]
\[x(x - 2) - 1(x - 2) = 0\]
\[(x - 2)(x - 1) = 0\]
\[x = 1,2\]
Hence, the points are \[(1,0),(2,0)\].
The correct option is D.

Additional information:
The point that lies on the x-axis is in the form (a,0), where a is a real number.
The point that lies on the y-axis is in the form (0,b), where b is a real number.

Note: In this particular type of question, the answer can also be given as the graph. To give the answer graphically one should know the process of obtaining a graph properly. First substitute 0, 1 , 2 etc. for x in the given equation to obtain the corresponding values of y to obtain the required points of the graph.