Recent content by debrajr

Find the equation of the conicoid $$2x^2-y^2=z^2+2x-7$$ when the origin is shifted to $$(2,-2,0)$$ and the axes are rotated so that the new axes have direction ratios $$-1,0,1;1,-2,1;0,1,1$$

Check whether the conicoid represented by $$3x^2-5y^2+z^2-6xy+7yz=15$$ is central or not. If it is central, obtain the center and the conics given by the intersection of the conicoid with the coordinate planes. If the given conicoid is not central, obtain all its tangent planes parallel to the...

The tangent and the normal to the conic $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ at a point $$(a\cos\left({\theta}\right), b\sin\left({\theta}\right))$$ meet the major axis in the points $$P$$ and $$P'$$, where $$PP'=a$$ Show that $$e^2cos^2\theta + cos\theta -1 = 0$$, where $$e$$ is the...

Prove that the paraboloids: $$\frac{x^2}{a_1^2}+\frac{y^2}{b_1^2}=\frac{2z}{c_1}$$; $$\frac{x^2}{a_2^2}+\frac{y^2}{b_2^2}=\frac{2z}{c_2}$$; $$\frac{x^2}{a_3^2}+\frac{y^2}{b_3^2}=\frac{2z}{c_3}$$ Have a common tangent plane if: $$\begin{bmatrix}a_1^2 & a_2^2 & a_3^2\\ b_1^2 & b_2^2 & b_3^2\\...