The tangents to `x^2+y^2=a^2` having inclinations `alpha` and `beta` intersect at `Pdot` If `cotalpha+cotbeta=0` , then find the locus of `Pdot`

If alpha,beta are roots of x^2-3x+a=0 , a in R and alpha <1< beta then find the value of a.

If we reduce 3x+3y+7=0 to the form xcosalpha+ysinalpha=p , then find the value of pdot

The tangent to y^(2)=ax make angles theta_(1)andtheta_(2) with the x-axis. If costheta_(1)costheta_(2)=lamda , then the locus of their point of intersection is

Tangents drawn from the point (c, d) to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 make angles alpha and beta with the x-axis. If tan alpha tan beta=1 , then find the value of c^(2)-d^(2) .

From an external point P , a pair of tangents is drawn to the parabola y^2=4xdot If theta_1a n dtheta_2 are the inclinations of these tangents with the x-axis such that theta_1+theta_2=pi/4 , then find the locus of Pdot

If a tangent to the parabola y^2=4a x meets the x-axis at T and intersects the tangents at vertex A at P , and rectangle T A P Q is completed, then find the locus of point Qdot

Let P be a point on the hyperbola x^2-y^2=a^2, where a is a parameter, such that P is nearest to the line y=2xdot Find the locus of Pdot

If the tangents are drawn to the circle x^2+y^2=12 at the point where it meets the circle x^2+y^2-5x+3y-2=0, then find the point of intersection of these tangents.

If the chord of contact of tangents from a point P to the parabola y^2=4a x touches the parabola x^2=4b y , then find the locus of Pdot

If the tangents to the parabola y^2=4a x intersect the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 at Aa n dB , then find the locus of the point of intersection of the tangents at Aa n dBdot