Prove that the square of the distance between the two points `(x_(1), y_(1))` and `(x_(2), y_(2))` of the circle `x^(2) + y^(2) = a^(2)` is `2(a^(2) - x_(1)x_(2) - y_(1)y_(2))`.

Find the area of the region enclosed between the two circles x^(2) +y^(2) =1 and (x-1)^(2) +y^(2)=1

Show that the equation of the chord of the parabola y^(2) = 4ax through the points (x_(1),y_(1)) and (x_(2),y_(2)) on it is (y-y_(1))(y-y_(2)) = y^(2) - 4ax

Find the lengths of the tangents drawn from the point. (x_(1),y_(1)) to the circle x^(2)+y^(2)+2x=0

Show that the equation of the straight line joining the points (x_(1),y_(1))and(x_(2),y_(2)) can be expressed in the form (x-x_(2))(y-y_(1))=(x-x_(1))(y-y_(2)) .

Three points P (h, k), Q(x_(1) , y_(1))" and " R (x_(2) , Y_(2)) lie on a line. Show that (h - x_(1)) (y_(2) - y_(1)) = (k - y_(1)) (x_(2) - x_(1)) .

The line y=mx+1 touches the curves y=-x^(4)+2x^(2)+x at two points P(x_(1),y_(1)) and Q(x_(2),y_(2)) . The value of x_(1)^(2)+x_(2)^(2)+y_(1)^(2)+y_(2)^(2) is

If a straight line passing through the focus of the parabola y^(2) = 4ax intersectts the parabola at the points (x_(1), y_(1)) and (x_(2), y_(2)) , then prove that x_(1)x_(2)=a^(2) .

Find the common tangents to the hyperbola x^(2)-2y^(2)=4 and the circle x^(2)+y^(2)=1

Prove that if P(x,y) be any point on the line segment joining P_(1)(x_(1),y_(1)) and P_(2) (x_(2),y_(2)) then x=x_(1)+k(x_(2)-x_(1)) and y=y_(1)+k(y_(2)-y_(1)) , where bar (P_(1)P), bar (P_(1)P_(2))=k What conclusion can you draw about the position of P if k=(1)/(2) ?

If a straight line pasing through the focus of the parabola y^(2) = 4ax intersects the parabola at the points (x_(1), y_(1)) and (x_(2) , y_(2)) then prove that y_(1)y_(2)+4x_(1)x_(2)=0 .