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find value of a when the distance between the points (3,a) and (4,1) is root 10.
Jun 30, 2024
find value of a when the distance between the points (3,a) and (4,1) is root 10.
Solution by Steps
step 1
Use the distance formula to find the distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2): (x2−x1)2+(y2−y1)2 \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
step 2
Substitute the given points (3,a)(3, a) and (4,1)(4, 1) into the distance formula: (4−3)2+(1−a)2=1+(1−a)2 \sqrt{(4 - 3)^2 + (1 - a)^2} = \sqrt{1 + (1 - a)^2}
step 3
Set the distance equal to 10\sqrt{10}: 1+(1−a)2=10 \sqrt{1 + (1 - a)^2} = \sqrt{10}
step 4
Square both sides to eliminate the square root: 1+(1−a)2=10 1 + (1 - a)^2 = 10
step 5
Simplify the equation: (1−a)2=9 (1 - a)^2 = 9
step 6
Take the square root of both sides: 1−a=±3 1 - a = \pm 3
step 7
Solve for aa: 1−a=3 1 - a = 3 or 1−a=−3 1 - a = -3
step 8
For 1−a=3 1 - a = 3 , a=−2 a = -2
step 9
For 1−a=−3 1 - a = -3 , a=4 a = 4
Answer
The value of aa can be either −2-2 or 44.
Key Concept
Distance formula
Explanation
The distance formula is used to find the distance between two points in a plane. By setting the distance equal to the given value and solving the resulting equation, we can find the possible values for aa.
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