Question

If the line $5y=7x$ divides the line segment joining the points $(3,5)$ and $(-5,2)$ in the ratio $m:n$ (where $\gcd(m,n)=1$), then the least value of $m^{2}+n^{2}$ is}

• A. 1675
• B. 2041
• C. 2185
• D. 2871

Hint:

For a line \(f(x,y)=0\) dividing the segment joining points \(P\) and \(Q\), directly use \[ \frac{m}{n} = -\frac{f(P)}{f(Q)}. \] Here, \[ f(3,5)=-4,\qquad f(-5,2)=-45, \] which immediately gives \[ m:n=4:45. \] Then simply compute \[ m^2+n^2=4^2+45^2=16+2025=2041. \] This method avoids using the section formula and saves significant time in competitive examinations.

Answer 1

The Correct Option is B

Concept: When a straight line divides the line segment joining two points, the ratio in which the segment is divided can be obtained using the values of the line equation at the two endpoints. If a line is represented by \[ f(x,y)=ax+by+c=0, \] and it divides the line segment joining points \[ P(x_1,y_1) \quad \text{and} \quad Q(x_2,y_2), \] then the ratio of division is given by \[ \frac{m}{n} = -\frac{f(P)}{f(Q)}. \] This result is extremely useful because it allows us to find the ratio directly without determining the exact point of intersection.

Step 1: Convert the given line into standard form. The given line is \[ 5y=7x. \] Bringing all terms to one side, \[ 7x-5y=0. \] Therefore, \[ f(x,y)=7x-5y. \]

Step 2: Evaluate the expression $f(x,y)$ at the first endpoint. The first endpoint of the segment is \[ P(3,5). \] Substituting these coordinates into the line equation: \[ f(3,5)=7(3)-5(5). \] \[ f(3,5)=21-25. \] \[ f(3,5)=-4. \] Thus, \[ f(P)=-4. \]

Step 3: Evaluate the expression $f(x,y)$ at the second endpoint. The second endpoint is \[ Q(-5,2). \] Substituting these coordinates: \[ f(-5,2)=7(-5)-5(2). \] \[ f(-5,2)=-35-10. \] \[ f(-5,2)=-45. \] Thus, \[ f(Q)=-45. \]

Step 4: Find the ratio in which the line divides the segment. Using the formula \[ \frac{m}{n} = -\frac{f(P)}{f(Q)}, \] we obtain \[ \frac{m}{n} = -\frac{-4}{-45}. \] \[ \frac{m}{n} = -\frac{4}{45}. \] The negative sign indicates that the division is external. Since only the magnitude of the ratio is required, \[ m:n = 4:45. \] Also, \[ \gcd(4,45)=1, \] so the ratio is already in its lowest terms. Hence, \[ m=4,\qquad n=45. \]

Step 5: Calculate the required value of $m^{2+n^{2}$.} Substituting the values of $m$ and $n$: \[ m^{2}+n^{2} = 4^{2}+45^{2}. \] Calculating individually, \[ 4^{2}=16, \] and \[ 45^{2}=2025. \] Therefore, \[ m^{2}+n^{2} = 16+2025. \] \[ m^{2}+n^{2} = 2041. \] Hence, the least value of \[ m^{2}+n^{2} \] is \[ \boxed{2041}. \]