A circle in the xy-plane has its center at the origin and has radius 5. The line y = 3/4x...

1. TRANSLATE the problem information

• Given information:
  • Circle centered at origin with radius 5
  • Line has equation \(\mathrm{y = \frac{3}{4}x}\)
  • Need positive x-coordinate where they intersect

• This tells us we need the circle equation \(\mathrm{x^2 + y^2 = 25}\)

2. INFER the solution approach

• To find intersection points, we need to solve the system of equations simultaneously
• The most efficient method is substitution - replace y in the circle equation with \(\mathrm{\frac{3}{4}x}\)

3. SIMPLIFY through substitution

• Substitute \(\mathrm{y = \frac{3}{4}x}\) into \(\mathrm{x^2 + y^2 = 25}\):
  \(\mathrm{x^2 + (\frac{3}{4}x)^2 = 25}\)

• Expand the squared term:
  \(\mathrm{x^2 + \frac{9}{16}x^2 = 25}\)

• Factor out \(\mathrm{x^2}\):
  \(\mathrm{x^2(1 + \frac{9}{16}) = 25}\)
  \(\mathrm{x^2(\frac{25}{16}) = 25}\)

• Solve for \(\mathrm{x^2}\):
  \(\mathrm{x^2 = 25 \times \frac{16}{25} = 16}\)

• Take the square root:
  \(\mathrm{x = ±4}\)

4. APPLY CONSTRAINTS to select final answer

• Since the problem asks for the positive x-coordinate: \(\mathrm{x = 4}\)

Answer: C) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(\frac{3}{4}x)^2}\) or combining like terms.

Common mistakes include writing \(\mathrm{(\frac{3}{4}x)^2 = \frac{3}{4}x^2}\) instead of \(\mathrm{\frac{9}{16}x^2}\), or incorrectly combining \(\mathrm{x^2 + \frac{9}{16}x^2}\) as \(\mathrm{\frac{10}{16}x^2}\) instead of \(\mathrm{\frac{25}{16}x^2}\). These calculation errors lead to wrong values for \(\mathrm{x^2}\) and ultimately incorrect intersection points.

This may lead them to select Choice B (3) or cause confusion and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students find \(\mathrm{x = ±4}\) correctly but fail to read carefully that the problem asks for the positive x-coordinate.

They might select the negative value or get confused about which root to choose, especially if they don't verify their answer by checking that both intersection points make sense.

This may lead them to select Choice A (-4).

The Bottom Line:

This problem combines algebraic manipulation with careful reading - students must execute the substitution accurately AND pay attention to the constraint about selecting the positive coordinate.