Question

Determine the distance OP.

Answer 1

Given points:
\(O = (0, 0)\), \(P = (8, 6)\)

Step 1: Use distance formula between two points
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Here,
\[ x_1 = 0, \quad y_1 = 0, \quad x_2 = 8, \quad y_2 = 6 \]

Step 2: Calculate distance \(OP\)
\[ OP = \sqrt{(8 - 0)^2 + (6 - 0)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

Final Answer:
\[ \boxed{10} \]

Answer 2

Given:
Equation of line \(QS\):
\[ 2x + 9y = 42 \]

Step 1: Find the coordinates where the line intersects the y-axis
At y-axis, \(x = 0\). Substitute \(x = 0\) into the equation:
\[ 2(0) + 9y = 42 \implies 9y = 42 \implies y = \frac{42}{9} = \frac{14}{3} \]

Final Answer:
The line \(QS\) intersects the y-axis at:
\[ \boxed{\left(0, \frac{14}{3}\right)} \]

Answer 3

Given:
- Points \(O = (0,0)\) and \(P = (8,6)\)
- Point \(R = (4.8, y)\) divides the line segment \(OP\) in some ratio \(k : 1\)

Step 1: Use section formula
If point \(R\) divides segment \(OP\) in ratio \(k : 1\), coordinates of \(R\) are:
\[ \left( \frac{k \times x_2 + 1 \times x_1}{k + 1}, \frac{k \times y_2 + 1 \times y_1}{k + 1} \right) \] Where \(O = (x_1, y_1) = (0,0)\) and \(P = (x_2, y_2) = (8,6)\).

Step 2: Set x-coordinate equal to 4.8 and solve for \(k\)
\[ 4.8 = \frac{8k + 0}{k + 1} = \frac{8k}{k + 1} \] Multiply both sides by \(k + 1\):
\[ 4.8(k + 1) = 8k \implies 4.8k + 4.8 = 8k \] \[ 8k - 4.8k = 4.8 \implies 3.2k = 4.8 \implies k = \frac{4.8}{3.2} = 1.5 \]

Step 3: Find \(y\) using \(k = 1.5\)
\[ y = \frac{6k + 0}{k + 1} = \frac{6 \times 1.5}{1.5 + 1} = \frac{9}{2.5} = 3.6 \]

Final Answer:
- The ratio in which \(R\) divides \(OP\) is \(1.5 : 1\) or \(\frac{3}{2} : 1\).
- The value of \(y = 3.6\).

\[ \boxed{ \text{Ratio} = \frac{3}{2} : 1, \quad y = 3.6 } \]

Answer 4

Given:
Points \(P\), \(Q\), \(O\), and \(S\) with coordinates:
- \(P = (8, 6)\)
- \(Q = (12, 2)\)
- \(O = (0, 0)\)
- \(S = (-6, 6)\)

Step 1: Calculate distance \(PQ\) using distance formula
\[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(12 - 8)^2 + (2 - 6)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4 \sqrt{2} \]

Step 2: Calculate distance \(OS\) using distance formula
\[ OS = \sqrt{(-6 - 0)^2 + (6 - 0)^2} = \sqrt{(-6)^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 6 \sqrt{2} \]

Step 3: Calculate the ratio \(\frac{PQ}{OS}\)
\[ \frac{PQ}{OS} = \frac{4 \sqrt{2}}{6 \sqrt{2}} = \frac{4}{6} = \frac{2}{3} \]

Final Answer:
\[ \boxed{\frac{PQ}{OS} = \frac{2}{3}} \]