The distance between the two points $P of open paren x sub 1 comma y sub 1 close paren$ and $R of open paren x sub 2 comma y sub 2 close paren$ is $d(P, R) = $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$ . Distance formula. Find the closest point on the line $y equals 2 x$ to the point $open paren 1 comma 7 close paren$ . (Hint: Every point on $y equals 2 x$ has the form $open paren x comma 2 x close paren$ , and the closest point has the minimum distance.)

검증된 단계별 안내

Identify the general form of any point on the line y = 2x, which is given as (x, 2x).

Write the distance formula between the point (1, 7) and a general point on the line (x, 2x): $d = \sqrt{(x - 1)^2 + (2x - 7)^2}$.

Since the square root function is increasing, minimize the squared distance instead to simplify calculations: $d^2 = (x - 1)^2 + (2x - 7)^2$.

Expand and simplify the expression for $d^2$ to get a quadratic function in terms of $x$.

Find the value of $x$ that minimizes $d^2$ by taking the derivative with respect to $x$, setting it equal to zero, and solving for $x$. Then, substitute this $x$ back into $y = 2x$ to find the corresponding $y$ coordinate of the closest point.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Distance Formula

The distance formula calculates the distance between two points in the coordinate plane using their coordinates. It is derived from the Pythagorean theorem and is given by d = √((x₁ - x₂)² + (y₁ - y₂)²). This formula helps quantify how far apart two points are.

Equation of a Line

The equation y = 2x represents a line where every point satisfies the relationship between x and y coordinates. Understanding this allows us to express any point on the line as (x, 2x), which is essential for substituting into the distance formula to find the closest point.

Minimizing Distance Using Calculus or Algebra

To find the closest point on a line to a given point, we minimize the distance function with respect to x. This involves expressing distance as a function of x, then finding its minimum value by setting the derivative to zero or using algebraic methods, ensuring the shortest distance is found.