Question

If the equation : \( (a^2+b^2)x^2 - 2(ac+bd)x + (c^2+d^2) = 0 \) has equal roots then :

• A. \(ab = cd\)
• B. \(ad = bc\)
• C. \(ad = \sqrt{bc}\)
• D. \(ab = \sqrt{cd}\)

Hint:

Condition for equal roots of \(Ax^2+Bx+C=0\) is Discriminant \(D = B^2-4AC=0\). Here, \(A=(a^2+b^2)\), \(B=-2(ac+bd)\), \(C=(c^2+d^2)\). \(B^2 = 4(ac+bd)^2 = 4(a^2c^2+b^2d^2+2abcd)\). \(4AC = 4(a^2+b^2)(c^2+d^2) = 4(a^2c^2+a^2d^2+b^2c^2+b^2d^2)\). Set \(B^2 = 4AC\) (since \(B^2-4AC=0\)): \(4(a^2c^2+b^2d^2+2abcd) = 4(a^2c^2+a^2d^2+b^2c^2+b^2d^2)\). Divide by 4 and simplify: \(a^2c^2+b^2d^2+2abcd = a^2c^2+a^2d^2+b^2c^2+b^2d^2\). \(2abcd = a^2d^2+b^2c^2\). Rearrange: \(a^2d^2 - 2abcd + b^2c^2 = 0\). This is \((ad-bc)^2 = 0\). So, \(ad-bc=0 \implies ad=bc\).

Answer 1

The Correct Option is B

Concept: For a quadratic equation \(Ax^2 + Bx + C = 0\), the roots are equal if and only if its discriminant (\(D\)) is equal to zero. The discriminant is given by \(D = B^2 - 4AC\). Step 1: Identify A, B, and C from the given quadratic equation The given equation is \( (a^2+b^2)x^2 - 2(ac+bd)x + (c^2+d^2) = 0 \). Comparing with \(Ax^2 + Bx + C = 0\):
\(A = a^2+b^2\)
\(B = -2(ac+bd)\)
\(C = c^2+d^2\) Step 2: Set the discriminant \(D = B^2 - 4AC\) to zero Since the equation has equal roots, \(D=0\). \[ [-2(ac+bd)]^2 - 4(a^2+b^2)(c^2+d^2) = 0 \] Step 3: Expand and simplify the equation \[ 4(ac+bd)^2 - 4(a^2+b^2)(c^2+d^2) = 0 \] Divide the entire equation by 4: \[ (ac+bd)^2 - (a^2+b^2)(c^2+d^2) = 0 \] Expand \((ac+bd)^2\): \( (ac+bd)^2 = (ac)^2 + (bd)^2 + 2(ac)(bd) = a^2c^2 + b^2d^2 + 2abcd \) Expand \((a^2+b^2)(c^2+d^2)\): \( (a^2+b^2)(c^2+d^2) = a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \) Now substitute these back into the equation: \[ (a^2c^2 + b^2d^2 + 2abcd) - (a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2) = 0 \] \[ a^2c^2 + b^2d^2 + 2abcd - a^2c^2 - a^2d^2 - b^2c^2 - b^2d^2 = 0 \] Cancel out terms: \(a^2c^2\) cancels with \(-a^2c^2\), and \(b^2d^2\) cancels with \(-b^2d^2\). We are left with: \[ 2abcd - a^2d^2 - b^2c^2 = 0 \] Multiply by -1 to make the squared terms positive (optional, but helps in recognizing a pattern): \[ a^2d^2 + b^2c^2 - 2abcd = 0 \] Step 4: Factor the resulting expression The expression \(a^2d^2 - 2abcd + b^2c^2\) is a perfect square trinomial. It can be written as \((ad)^2 - 2(ad)(bc) + (bc)^2\). This is of the form \((X-Y)^2 = X^2 - 2XY + Y^2\), where \(X=ad\) and \(Y=bc\). So, \((ad - bc)^2 = 0\). Step 5: Solve for the condition If \((ad - bc)^2 = 0\), then taking the square root of both sides gives: \[ ad - bc = 0 \] \[ ad = bc \] This is the condition for equal roots.