Answer each question. Does the straight line 3x - 2y = 9 intersect the circle x2 + y2 = 25? (Hint: To find out, solve the system formed by these two equations.)

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 63

6장, 문제 63

Answer each question. Does the straight line 3x - 2y = 9 intersect the circle x² + y² = 25? (Hint: To find out, solve the system formed by these two equations.)

검증된 단계별 안내

Rewrite the equation of the line $3x - 2y = 9$ to express $y$ in terms of $x$. Start by isolating $y$: $3x - 2y = 9$ becomes $-2y = 9 - 3x$, then $y = \frac{3x - 9}{2}$.

Substitute the expression for $y$ from the line equation into the circle equation $x^2 + y^2 = 25$. This gives $x^2 + \left(\frac{3x - 9}{2}\right)^2 = 25$.

Simplify the equation by expanding the squared term and multiplying through by 4 to clear the denominator: $4x^2 + (3x - 9)^2 = 100$.

Expand $(3x - 9)^2$ to get $9x^2 - 54x + 81$, then combine like terms with \$4x^2$ to form a quadratic equation in $x$: $4x^2 + 9x^2 - 54x + 81 = 100$.

Bring all terms to one side to set the quadratic equal to zero: $13x^2 - 54x + 81 - 100 = 0$, which simplifies to $13x^2 - 54x - 19 = 0$. Analyze the discriminant $\Delta = b^2 - 4ac$ to determine if there are real solutions, indicating intersection points.

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주요 개념

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

Equation of a Straight Line

A straight line in the plane can be represented by a linear equation like 3x - 2y = 9. Understanding how to manipulate and solve such equations is essential for finding points of intersection with other curves.

Equation of a Circle

A circle centered at the origin with radius r is described by the equation x² + y² = r². Recognizing this form helps in setting up systems of equations to find where a line might intersect the circle.

Solving Systems of Equations

To determine if the line intersects the circle, solve the system formed by their equations simultaneously. This involves substitution or elimination methods to find common solutions (x, y) that satisfy both equations.