Convert each equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci. 4x² + y²+ 16x - 6y - 39 = 0

검증된 단계별 안내

Start with the given equation: \(4x^{2} + y^{2} + 16x - 6y - 39 = 0\).

Group the \(x\) terms and \(y\) terms together: \(4x^{2} + 16x + y^{2} - 6y = 39\).

Factor out the coefficient of \(x^{2}\) from the \(x\) terms: \(4(x^{2} + 4x) + (y^{2} - 6y) = 39\).

Complete the square for both \(x\) and \(y\) terms inside the parentheses: - For \(x^{2} + 4x\), take half of 4 (which is 2), square it (which is 4), and add inside the parentheses. - For \(y^{2} - 6y\), take half of -6 (which is -3), square it (which is 9), and add inside the parentheses.

Adjust the equation by adding the equivalent values outside the parentheses to keep the equation balanced, then rewrite the equation in standard form of an ellipse: \(rac{(x + h)^{2}}{a^{2}} + rac{(y + k)^{2}}{b^{2}} = 1\), where \((h, k)\) is the center.

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주요 개념

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

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)² or (y - k)² by adding and subtracting terms. This technique helps transform the given equation into a standard form, making it easier to identify the center and dimensions of conic sections like ellipses.

Standard Form of an Ellipse

The standard form of an ellipse equation is (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center, and a and b are the lengths of the semi-major and semi-minor axes. Converting to this form allows for straightforward graphing and analysis of the ellipse's properties.

Foci of an Ellipse

The foci are two fixed points inside an ellipse such that the sum of the distances from any point on the ellipse to the foci is constant. Their locations depend on the values of a, b, and the center, and are found using the relationship c² = |a² - b²|, where c is the distance from the center to each focus.