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Magazine Chapter 19: Problem 60
The point on the straight line \(y=2 x+11\) which is nearest to the circle \(16\left(x^{2}+y^{2}\right)+32 x-8 y-50=0\) is (A) \(\left(\frac{9}{2}, 2\right)\) (B) \(\left(-\frac{9}{2}, 2\right)\) (C) \(\left(\frac{9}{2},-2\right)\) (D) none of these
Short Answer
Expert verified
(B) \((-\frac{9}{2}, 2)\)
Step by step solution
01
Rewrite the Circle Equation
The given circle equation is \(16(x^2 + y^2) + 32x - 8y - 50 = 0\). First, let's rewrite this equation by rearranging and simplifying using algebraic methods. Divide the entire equation by 16 to simplify it:\[x^2 + y^2 + 2x - \frac{1}{2}y - \frac{25}{8} = 0\]
02
Complete the Square
Next, complete the square for the terms involving \(x\) and \(y\) to convert the circle equation into standard form. Starting with \(x\) terms:- Take the term \(x^2 + 2x\). - To complete the square, add and subtract \((\frac{2}{2})^2 = 1\): \[x^2 + 2x = (x+1)^2 - 1\]Similarly, for \(y\) terms, take \(y^2 - \frac{1}{2}y\):- Add and subtract \((\frac{-1/2}{2})^2 = \frac{1}{16}\): \[y^2 - \frac{1}{2}y = (y - \frac{1}{4})^2 - \frac{1}{16}\]
03
Form the Circle's Standard Equation
Substitute the completed square terms back into the equation and rearrange:\[(x+1)^2 - 1 + (y - \frac{1}{4})^2 - \frac{1}{16} - \frac{25}{8} = 0\]Combine constants to simplify:\[(x+1)^2 + (y - \frac{1}{4})^2 = \frac{81}{16}\]This is now a standard form of a circle equation with center \((-1, \frac{1}{4})\) and radius \(\frac{9}{4}\).
04
Express Line in Parametric Form
The line is \(y = 2x + 11\). We can express a point \((x, y)\) on the line in terms of \(x\) alone:- Let \(x = t\). Then \(y = 2t + 11\).- So, a point on the line can be parameterized as \((t, 2t + 11)\).
05
Set Up Distance Formula
The task is to minimize the distance between a point \((t, 2t + 11)\) on the line and the circle's center \((-1, \frac{1}{4})\).The distance \(d\) between these two points is given by:\[d = \sqrt{(t + 1)^2 + (2t + 11 - \frac{1}{4})^2}\]
06
Calculate and Minimize Squared Distance
Minimizing \(d\) is the same as minimizing \(d^2\). Let's calculate \(d^2\):\[d^2 = (t + 1)^2 + (2t + \frac{43}{4})^2\]Expand and differentiate this expression with respect to \(t\) to find the minimum:
07
Differentiation and Solve for Critical Point
Expand \(d^2 = (t+1)^2 + (2t + \frac{43}{4})^2\), differentiate with respect to \(t\) and set the derivative to zero to find critical points. Then solve:\[\frac{d}{dt}\left((t+1)^2 + (2t+\frac{43}{4})^2\right) = 2(t+1) + 4(2t + \frac{43}{4}) = 0 \]Simplify and solve this linear equation to find \(t\).
08
Evaluate t and Find Nearest Point
Solve the equation from step 7:- Simplifying gives: \[2t + 2 + 8t + \frac{43}{2} = 0\] \[10t + \frac{47}{2} = 0\] \[10t = -\frac{47}{2}\] \[t = -\frac{47}{20}\]Substitute this back in the line equation \((t, 2t+11)\) for \(t\), and calculate: \[x = -\frac{47}{20}, \quad y = 2(-\frac{47}{20}) + 11 = 2\cdot (-\frac{47}{20}) + \frac{220}{20} = \frac{9}{2}\]Thus, the point is \(\left(-\frac{9}{2}, 2\right)\).
09
Conclusion
Based on calculations, the point on the line \(y = 2x + 11\) closest to the circle \((x+1)^2 + (y - \frac{1}{4})^2 = \frac{81}{16}\) is \((-\frac{9}{2}, 2)\), corresponding to option (B).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Circle Equation
The circle equation is a fundamental concept in coordinate geometry. It helps us to represent a circle on the Cartesian plane using an algebraic expression. The general form is \[Ax^2 + By^2 + Cx + Dy + E = 0\]which can be transformed into the standard form. The standard form, \[(x - h)^2 + (y - k)^2 = r^2\], is more insightful:
Understanding the different forms of the circle equation assists in solving various geometrical problems, like locating points or measuring distances.
- \((h, k)\) is the center of the circle.
- \(r\) is the radius.
Understanding the different forms of the circle equation assists in solving various geometrical problems, like locating points or measuring distances.
Parametric Form
The parametric form is a way of expressing the coordinates of points on geometric shapes using a parameter. In this context, converting the line equation \(y = 2x + 11\) into parametric form simplifies finding specific points on the line.
- Choose a parameter, such as \(t\) for \(x\), leading to \(x = t\).
- Substitute into the line equation: \(y = 2t + 11\).
- The line can then be expressed as points \((t, 2t+11)\).
Distance Formula
The distance formula provides a way to calculate the distance between two points in a plane. Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) is expressed as: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In problems involving geometry, such as finding the closest point on a line to a circle, this formula is key.
- Calculate the difference in \(x\) and \(y\) coordinates.
- Square these differences, sum them, and take the square root.
Completing the Square
Completing the square is a method used to transform quadratics into a perfect square trinomial. It is vital when converting the general circle equation into standard form. Here's the process:
- Identify the quadratic terms in \(x\) and \(y\).
- For each, add and subtract the square of half the linear coefficient to form a perfect square.
- This reduces expressions like \(x^2 + bx\) into \((x + \frac{b}{2})^2 - (\frac{b}{2})^2\).
