Two tangents to the parabola `y^2=8x` meet the tangent at the vertex in the point P and Q. If `PQ=4` then prove that the equation of the locus of the point of intersection of two tangents is `y^2=8(x+2)`

Two tangents to the hyperbola x^2/a^2-y^2/b^2=1 make angles theta_1,theta_2 , with the transverse axis. Find the locus of their point of intersection if tantheta_1+tantheta_2=k .

If tangents are drawn to the circle x^2 +y^2=12 at the points where it intersects the circle x^2+y^2-5x+3y-2=0 , then the coordinates of the point of intersection of those tangents are

A tangent to the parabola y^2=8x makes an angle of 45^0 with the straight line y=3x+5. Then find one of the points of contact.

P and Q are two points on the ellipse x^2/a^2+y^2/b^2=1 with eccentric angles theta_1 and theta_2 . Find the equation of the locus of the point of intersection of the tangents at P and Q if theta_1+theta_2=pi/2 .

The vertex of the parabola y^2-3x+8y+31=0 , is

The tangent to the curve y=e^(2x) at the point (0,1) meets X-axis at

The tangent to the curve y=e^(2x) at the point (0,1) meets X-axis at

the equation of the tangent to the circle x^2+y^2=17 at the point (1,-4) is

Find the equation of the tangent to the curve y=x-sinxcosx at x=pi/2