Question

Suppose a line parallel to \(ax+by=0\) (where \(b≠0\))intersects\( 5x-y+4=0\) and \(3x+4y-4=0\) ,respectively at P and Q. If the midpoint of PQ is \((1,5)\),then the value of \(\dfrac{a}{b}\) is

• A. \(\dfrac{107}{3}\)
• B. \(\dfrac{-107}{3}\)
• C. \(\dfrac{3}{107}\)
• D. \(\dfrac{-3}{107}\)
• E. \(1\)

Answer 1

The Correct Option is B

Given data

The given lines are:

\(5x - y + 4 = 0 ⇒ y = 5x + 4\)-----------(1)

\(3x + 4y - 4 = 0 ⇒ y = -(3/4)x + 1\)------(2)

From the equations, we can see that the slopes of the two lines are \(5\) and \(\dfrac{-3}{4}\), respectively.

Now, let the coordinates of point P be (p, 5p + 4) (since it lies on the line y = 5x + 4) and the coordinates of point Q be \((q, -(\dfrac{3}{4})q + 1) \)(since it lies on the line

 \(y = \dfrac{-3x}{4} + 1\)

Step 2:

 Find the midpoint of PQ. The midpoint of two points \((x1, y1) \)and \((x2, y2)\) is given by: 

Midpoint = \((\dfrac{(x1 + x2)}{2}, \dfrac{(y1 + y2)}{2})\)

Given that the midpoint of PQ is \((1, 5)\), we have: \((1, 5) = ((p + q)/2, ((5p + 4) - (3/4)q + 1)/2)\)

Now, we can write two equations based on the coordinates of the midpoint:

1. \((p + q)/2 = 1\)
2. \(((5p + 4) - \dfrac{(\dfrac{3}{4})q + 1)}{2} = 5\)

Step 3: 

Solve the system of equations to find p and q. From equation 1, we get: \(p + q = 2\)

From equation 2, we can simplify and \(get: 5p - (3/4)q + 5 = 10 5p - (3/4)q = 5\)

Now, let's multiply the second equation by 4 to get : \(20p - 3q = 20\)

Now, we can use the first equation \((p + q = 2)\) to solve for \(q: q = 2 - p\)

Substitute this value of \(q\) into the equation \((20p - 3q = 20): 20p - 3(2 - p) = 20 20p - 6 + 3p = 20 23p = 26 p = \dfrac{26}{23}\)

Now that we have the value of p, we can find the value of q:\( q = 2 - \dfrac{26}{23} q = \dfrac{46}{23}\)

Step 4:

 Find the equation of the line passing through P and Q. The slope of the line passing through P and Q is given by: 

\(m\) \(= \dfrac{(y2 - y1)}{(x2 - x1)}\)

Using the coordinates of \(P (p, 5p + 4)\) and \(Q (q, -(\dfrac{3}{4})q + 1)\), 

we get: \(m = (\dfrac{(-(3/4)}q + 1) - \dfrac{(5p + 4)}{(q - p)}\)

Now, substitute the values of p and q:

\( m= ((-(3/4)(46/23) + 1) - (5(26/23) + 4)) / ((46/23) - (26/23))\)

\(m= ((-69/92 + 1) - (130/23 + 92/23)) / (20/23)\)

 \(m = ((23/92 - 222/23) / (20/23)\)

\( m= ((23 - 5064) / 920) / (20/23) \)

\(m = (-5041/920) / (20/23)\)

\(m = -5041/920 * 23/20 \)

\(m = \dfrac{-107}{3}\) (_Ans)