Question

If m:n is the ratio in which the point $\left(\frac{8}{5}, \frac{1}{5}, \frac{8}{5}\right)$ divides the line segment joining the points (2,p,2) and (p,-2,p) where p is an integer then $\frac{3m+n}{3n}=$

• A. p
• B. 2p
• C. 3p
• D. 4p

Hint:

When faced with a problem in 3D coordinate geometry that seems to have inconsistent numbers (e.g., leads to an equation with no solution), carefully re-read the problem and double-check your application of formulas like the section formula. If the inconsistency persists, it's highly likely the problem statement itself contains errors.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The problem involves the section formula in three-dimensional geometry. We are given the coordinates of two endpoints, the point of division, and the ratio in symbolic form. We need to find the value of an integer parameter $p$ and then an expression involving the ratio.
Step 2: Key Formula or Approach
1. Let the dividing point be $P(\frac{8}{5}, \frac{1}{5}, \frac{8}{5})$, and the endpoints be $A(2,p,2)$ and $B(p,-2,p)$. The ratio is $m:n$. 2. Apply the section formula: $P = \frac{mB+nA}{m+n}$. 3. This gives three equations, one for each coordinate. Use these equations to find the integer $p$ and the ratio $m/n$. 4. Calculate the required expression.
Step 3: Detailed Explanation
The problem as stated in the source document leads to a contradiction. As shown by attempting to solve for $p$ using the section formula for all three coordinates, the resulting quadratic equation $5p^2 - 9p + 6 = 0$ has no real solutions for $p$, because its discriminant is $D = (-9)^2 - 4(5)(6) = 81 - 120 = -39<0$. This indicates a typo in the coordinates of the given points. Let's assume there is a typo in the y-coordinate of the first point A and it should be A(2, -p, 2). Let's solve with this correction. The points are now $A(2,-p,2)$ and $B(p,-2,p)$. The dividing point is $P(\frac{8}{5}, \frac{1}{5}, \frac{8}{5})$. Using the section formula for the y-coordinate: \[ \frac{m(-2) + n(-p)}{m+n} = \frac{1}{5} \] \[ 5(-2m-np) = m+n \implies -10m-5np = m+n \implies -11m = n(1+5p) \implies \frac{m}{n} = -\frac{1+5p}{11} \] Using the section formula for the x-coordinate (which is the same as for z): \[ \frac{m(p) + n(2)}{m+n} = \frac{8}{5} \] \[ 5(mp+2n) = 8(m+n) \implies 5mp+10n = 8m+8n \implies m(5p-8) = -2n \implies \frac{m}{n} = -\frac{2}{5p-8} \] Equating the expressions for $m/n$: \[ -\frac{1+5p}{11} = -\frac{2}{5p-8} \] \[ (1+5p)(5p-8) = 22 \] \[ 25p^2 - 40p + 5p - 8 = 22 \] \[ 25p^2 - 35p - 30 = 0 \] Divide by 5: \[ 5p^2 - 7p - 6 = 0 \] Factor the quadratic: $(5p+3)(p-2)=0$. The possible values for $p$ are $p=2$ or $p=-3/5$. Since $p$ is an integer, we must have $p=2$. Now find the ratio $m/n$ using $p=2$: \[ \frac{m}{n} = -\frac{2}{5p-8} = -\frac{2}{5(2)-8} = -\frac{2}{10-8} = -\frac{2}{2} = -1 \] So the ratio is $m:n = -1:1$, which means P is the midpoint. Let's check: Midpoint = $(\frac{2+p}{2}, \frac{-p-2}{2}, \frac{2+p}{2})$. With $p=2$, midpoint is $(\frac{4}{2}, \frac{-4}{2}, \frac{4}{2})=(2,-2,2)$. This does not match P. So the section formula with ratio is necessary. The ratio is $m/n=-1$. Now we calculate the required expression: \[ \frac{3m+n}{3n} = \frac{3(m/n)+1}{3} = \frac{3(-1)+1}{3} = \frac{-2}{3} \] This is a constant, while the options involve $p$. This indicates the correction might be wrong, or the options are wrong. Given the high likelihood of typos, there is no certain path. However, let's assume the option `p` is correct and work backwards to find a consistent set of numbers. This is not a valid solving method. We must conclude the question is flawed. Step 4: Final Answer
The question as stated is inconsistent and has no solution. The data leads to a quadratic equation for the integer $p$ which has no real roots. Correcting a potential typo does not lead to any of the given options. Thus, the problem is unsolvable.