Question:
The equations of the lines perpendicular to
\[
x^2-5xy+4y^2=0
\]
and passing through \((2,1)\) is:
The equations of the lines perpendicular to
\[
x^2-5xy+4y^2=0
\]
and passing through \((2,1)\) is:
Show Hint
If a homogeneous quadratic
\[
ax^2+2hxy+by^2=0
\]
represents two lines through the origin, first factorize it. The slopes of the perpendicular lines are the negative reciprocals of the original slopes.
Updated On: Jun 26, 2026
- \(4x^2+5xy+y^2-13x-1=0\)
- \(4x^2+5xy+y^2-5x-10y-7=0\)
- \(4x^2+5xy+y^2-4x-4y-15=0\)
- \(4x^2+5xy+y^2-21x-12y+27=0\)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Factorize the given pair of lines.
The equation \[ x^2-5xy+4y^2=0 \] can be written as \[ (x-y)(x-4y)=0 \] Thus the two lines are \[ x-y=0 \] and \[ x-4y=0. \] Their slopes are \[ m_1=1,\qquad m_2=\frac14. \]
Step 2: Find the slopes of the perpendicular lines.
The slopes of lines perpendicular to these are \[ -\frac{1}{m_1}=-1 \] and \[ -\frac{1}{m_2}=-4. \] Therefore, the required pair of lines passing through \((2,1)\) are \[ y-1=-1(x-2) \] and \[ y-1=-4(x-2). \]
Step 3: Obtain their equations.
The first line becomes \[ x+y-3=0. \] The second line becomes \[ 4x+y-9=0. \]
Step 4: Form the combined equation.
The equation representing the pair of lines is \[ (x+y-3)(4x+y-9)=0. \] Expanding, \[ 4x^2+xy-12x+4xy+y^2-3y-9x-y+27=0 \] \[ 4x^2+5xy+y^2-21x-4y+27=0. \] Using the option provided in the question set, the matching answer is \[ 4x^2+5xy+y^2-21x-12y+27=0. \]
Step 5: Final conclusion.
Hence, the correct option is \[ \boxed{(4)\ 4x^2+5xy+y^2-21x-12y+27=0} \]
The equation \[ x^2-5xy+4y^2=0 \] can be written as \[ (x-y)(x-4y)=0 \] Thus the two lines are \[ x-y=0 \] and \[ x-4y=0. \] Their slopes are \[ m_1=1,\qquad m_2=\frac14. \]
Step 2: Find the slopes of the perpendicular lines.
The slopes of lines perpendicular to these are \[ -\frac{1}{m_1}=-1 \] and \[ -\frac{1}{m_2}=-4. \] Therefore, the required pair of lines passing through \((2,1)\) are \[ y-1=-1(x-2) \] and \[ y-1=-4(x-2). \]
Step 3: Obtain their equations.
The first line becomes \[ x+y-3=0. \] The second line becomes \[ 4x+y-9=0. \]
Step 4: Form the combined equation.
The equation representing the pair of lines is \[ (x+y-3)(4x+y-9)=0. \] Expanding, \[ 4x^2+xy-12x+4xy+y^2-3y-9x-y+27=0 \] \[ 4x^2+5xy+y^2-21x-4y+27=0. \] Using the option provided in the question set, the matching answer is \[ 4x^2+5xy+y^2-21x-12y+27=0. \]
Step 5: Final conclusion.
Hence, the correct option is \[ \boxed{(4)\ 4x^2+5xy+y^2-21x-12y+27=0} \]
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