Question:
The midpoint of the diagonal of a rectangle formed by \(x^2+5x-6=0\) and \(y^2-8y-20=0\) is
The midpoint of the diagonal of a rectangle formed by \(x^2+5x-6=0\) and \(y^2-8y-20=0\) is
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If equations in \(x\) and \(y\) separately are given, first find their roots. The roots represent parallel lines, whose intersections form the vertices of the rectangle.
Updated On: Jun 25, 2026
- \(\left(\frac{5}{2},4\right)\)
- \(\left(-\frac{5}{2},-6\right)\)
- \(\left(-\frac{5}{2},4\right)\)
- \(\left(\frac{5}{2},-6\right)\)
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The Correct Option is C
Solution and Explanation
Step 1: Find the vertical sides of the rectangle.
Given: \[ x^2+5x-6=0 \] Factorizing, \[ (x+6)(x-1)=0 \] Thus, \[ x=-6 \quad \text{or} \quad x=1 \] These represent two vertical lines forming opposite sides of the rectangle.
Step 2: Find the horizontal sides of the rectangle.
Given: \[ y^2-8y-20=0 \] Factorizing, \[ (y-10)(y+2)=0 \] Thus, \[ y=10 \quad \text{or} \quad y=-2 \] These represent two horizontal lines forming the other pair of sides.
Step 3: Determine the vertices of the rectangle.
The rectangle is formed by the lines: \[ x=-6,\quad x=1,\quad y=10,\quad y=-2 \] Hence, the vertices are \[ (-6,10), \quad (1,10), \quad (-6,-2), \quad (1,-2) \]
Step 4: Find the midpoint of a diagonal.
Take the diagonal joining \[ (-6,10) \] and \[ (1,-2) \] Using midpoint formula: \[ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \] Substituting values: \[ \left( \frac{-6+1}{2}, \frac{10+(-2)}{2} \right) \] \[ = \left( \frac{-5}{2}, \frac{8}{2} \right) \] \[ = \left( -\frac{5}{2}, 4 \right) \]
Step 5: Final conclusion.
Therefore, the midpoint of the diagonal is \[ \boxed{\left(-\frac{5}{2},4\right)} \]
Given: \[ x^2+5x-6=0 \] Factorizing, \[ (x+6)(x-1)=0 \] Thus, \[ x=-6 \quad \text{or} \quad x=1 \] These represent two vertical lines forming opposite sides of the rectangle.
Step 2: Find the horizontal sides of the rectangle.
Given: \[ y^2-8y-20=0 \] Factorizing, \[ (y-10)(y+2)=0 \] Thus, \[ y=10 \quad \text{or} \quad y=-2 \] These represent two horizontal lines forming the other pair of sides.
Step 3: Determine the vertices of the rectangle.
The rectangle is formed by the lines: \[ x=-6,\quad x=1,\quad y=10,\quad y=-2 \] Hence, the vertices are \[ (-6,10), \quad (1,10), \quad (-6,-2), \quad (1,-2) \]
Step 4: Find the midpoint of a diagonal.
Take the diagonal joining \[ (-6,10) \] and \[ (1,-2) \] Using midpoint formula: \[ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \] Substituting values: \[ \left( \frac{-6+1}{2}, \frac{10+(-2)}{2} \right) \] \[ = \left( \frac{-5}{2}, \frac{8}{2} \right) \] \[ = \left( -\frac{5}{2}, 4 \right) \]
Step 5: Final conclusion.
Therefore, the midpoint of the diagonal is \[ \boxed{\left(-\frac{5}{2},4\right)} \]
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