Show that `xcosalpha+ysinalpha=p` touches the parabola `y^2=4a x` if `pcosalpha+asin^2alpha=0` and that the point of contact is `(atan^2alpha,-2atanalpha)dot`

Show that the straight line xcosalpha+ysinalpha=p touches the curve x y=a^2, if p^2=4a^2cosalphasinalphadot

Show that the line x + y + 1=0 touches the hyperbola x^(2)/16 -y^(2)/15 = 1 and find the co-ordinates of the point of contact.

The line x - y + 2 = 0 touches the parabola y^2 = 8x at the point

Find the locus of the midpoint of chords of the parabola y^2=4a x that pass through the point (3a ,a)dot

The slope of the line touching both the parabolas y^2=4x and x^2=−32y is

If the line 2x+sqrt6y=2 touches the hyperbola x^2-2y^2=4 , then the point of contact is

y=x is tangent to the parabola y=ax^(2)+c . If c=2, then the point of contact is

The equation to the line touching both the parabolas y^2 =4x and x^2=-32y is

If the straight line xcosalpha+ysinalpha=p touches the curve (x^2)/(a^2)+(y^2)/(b^2)=1 , then prove that a^2cos^2alpha+b^2sin^2alpha=p^2dot

If two tangents drawn from the point (alpha,beta) to the parabola y^2=4x are such that the slope of one tangent is double of the other, then prove that alpha=2/9beta^2dot