5 votes
Accepted

Determine the angles of quadrilateral that make it concyclic

COMMENT.-It is easy if we take into account that the opposite to the angle in a vertex must be its suplement. In fact $$\overline{DB}^2=a^2+d^2-2ad\cos(\phi)=b^2+c^2-2bc\cos(\pi-\phi)=b^2+c^2+2bc\cos(\...
Piquito's user avatar
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3 votes

find the strictly positive real number r in this equation with complex numbers

In polar form, the two conditions mean that $$ (r^2\cos2\theta+1)^2+(r^2\sin2\theta)^2=4r^2. $$ That is, $$ r^4-4r^2+1=-2r^2\cos2\theta. $$ Therefore, a solution $z=r(\cos\theta+i\sin\theta)$ exists ...
user1551's user avatar
  • 138k
1 vote

solve for t that minimizes/maximizes z(t) within an ellipsoidal cone

I would approach using Lagrange multiplier method. Let the vector $k = [0,0,1]^T$, and let $ r = [x,y,z]^T $, then you cone and ellipsoid equations are $ r^T Q_c r = 0 $ (for the cone), and $r^T Q_e ...
of course's user avatar
  • 20.8k
1 vote

Find the maximum value of $a$ under the condition $2e\ln x\leq ax+b\leq \frac{1}{2}x^2+e$,where $a,b\in\mathbb{R}$,$x>0$

When ax+b is tangent to the other two functions and b is the smallest, the value of a is the largest Let the tangent point be(x1,y1)(x2,y2)So the derivatives of the other functions is equal and equals ...
XHZ's user avatar
  • 11
1 vote
Accepted

find the strictly positive real number r in this equation with complex numbers

A solution based on analytic geometry: Render the equations as $|z^2+1|=2|z|$ Square both sides and express the squared norms in terms of real and imaginary parts: $(x^2-y^2+1)^2+4x^2y^2=4(x^2+y^2)$ $(...
Oscar Lanzi's user avatar
  • 38.7k

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