If the equation $(m^2+n^2)x^2-2(mp+nq)x+p^2+q^2=0$ has equal roots , then
- $mp = nq$
- $mq =np$
- $mn=pq$
- $mq=\sqrt{np}$
Reveal answer
Fill a bubble to check yourself
B
Correct answer
Explanation
For equal roots, the discriminant D = 0. D = 4(mp+nq)^2 - 4(m^2+n^2)(p^2+q^2) = 0. Expanding gives m^2p^2 + n^2q^2 + 2mpnq - (m^2p^2 + m^2q^2 + n^2p^2 + n^2q^2) = 0, which simplifies to -(mq - np)^2 = 0, so mq = np.
AI explanation
For the quadratic equation (m^2 + n^2)x^2 - 2(mp + nq)x + p^2 + q^2 = 0 to have equal roots, its discriminant must be zero. The discriminant is D = [-2(mp + nq)]^2 - 4(m^2 + n^2)(p^2 + q^2) = 0, which simplifies to 4(m^2p^2 + n^2q^2 + 2mnpq) - 4(m^2p^2 + m^2q^2 + n^2p^2 + n^2q^2) = 0. Dividing by -4 yields m^2q^2 + n^2p^2 - 2mnpq = 0, which factors as a perfect square (mq - np)^2 = 0. Therefore, mq = np.