Examkin

Question

Difficulty: HardDistance and Midpoint Formulas

In the standard (x,y)(x, y) coordinate plane, a line segment ABAB has midpoint M(3,4)M(3, 4). If endpoint AA lies on the line y=2x−7y = 2x - 7, and the length of segment ABAB is 1010 units, what is the product of all possible xx-coordinates of endpoint AA?

Answer: 21

Answer

The product of all possible xx-coordinates of endpoint AA is 2121.
By determining that the distance from endpoint A(x,2x−7)A(x, 2x - 7) to the midpoint M(3,4)M(3, 4) is half of the segment length ABAB (which is 55), we set up the distance formula equation: (x−3)2+(2x−11)2=25(x - 3)^2 + (2x - 11)^2 = 25. Simplifying this equation leads to the quadratic expression x2−10x+21=0x^2 - 10x + 21 = 0, which factors into (x−3)(x−7)=0(x - 3)(x - 7) = 0. The two possible xx-coordinates are 33 and 77. Multiplying these values yields the product 2121.

Step-by-Step Solution

1
Find the distance between endpoint AA and midpoint MM.
The distance AMAM is 55.
Since MM is the midpoint of segment ABAB of length 1010, the distance from either endpoint to the midpoint is half of the total length: 10÷2=510 \div 2 = 5.
2
Express the coordinates of endpoint AA in terms of a single variable.
Endpoint AA is represented as (x,2x−7)(x, 2x - 7).
Endpoint AA lies on the line y=2x−7y = 2x - 7.
3
Apply the distance formula to find the relationship for xx.
(x−3)2+(2x−11)2=25(x - 3)^2 + (2x - 11)^2 = 25
The distance between A(x,2x−7)A(x, 2x - 7) and M(3,4)M(3, 4) is 55, so the square of the distance is 52=255^2 = 25.
4
Simplify the quadratic equation.
x2−10x+21=0x^2 - 10x + 21 = 0
Expanding (x−3)2+(2x−11)2=25(x - 3)^2 + (2x - 11)^2 = 25 yields x2−6x+9+4x2−44x+121=25x^2 - 6x + 9 + 4x^2 - 44x + 121 = 25. Combining like terms gives 5x2−50x+130=255x^2 - 50x + 130 = 25. Subtracting 2525 from both sides gives 5x2−50x+105=05x^2 - 50x + 105 = 0. Dividing the entire equation by 55 yields x2−10x+21=0x^2 - 10x + 21 = 0.
5
Solve for the possible values of xx.
x=3x = 3 or x=7x = 7
Factoring the quadratic equation yields (x−3)(x−7)=0(x - 3)(x - 7) = 0, which gives the roots x=3x = 3 and x=7x = 7.
6
Calculate the product of the possible xx-coordinates.
21
The product of the two possible xx-coordinates is 3×7=213 \times 7 = 21.

Key Concept

Distance and Midpoint Formulas

Alternative Method

Instead of expanding the quadratic equation, one can use the geometric interpretation. The points AA are the intersections of the circle (x−3)2+(y−4)2=25(x - 3)^2 + (y - 4)^2 = 25 and the line y=2x−7y = 2x - 7. Substituting y=2x−7y = 2x - 7 directly into the circle equation and simplifying to x2−10x+21=0x^2 - 10x + 21 = 0 is the most direct approach.
Estimated Time:2m 0s
Rate this question

Topics

Distance and Midpoint Formulas

More questions from this topic