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))`.

Show that the equation of the chord of the parabola x^(2) = 4ay joining the points (x_(1),y_(1)) ann (x_(2),y_(2)) on it is (x-x_(1))(x-x_(2))=x^(2)-4ay .

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

The equation to the chord joining two points (x_1,y_1)a n d(x_2,y_2) on the rectangular hyperbola x y=c^2 is:

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)) .

A(x_(1) y_(1)) and B(x_(2), y_(2)) are two given points on a circle. If the chord AB substends an angle theta at a point p on its circumferene, show that the equation of the circle is (x-x_(1)) (x-x_(2))+(y-y_(1))(y-y_(2))=+- cos theta [ (x-x_(1))(y-y_(2))-(x-x_(2))(y-y_(1))]

The straight line joining the points p(x_(1),y_(1))andQ(x_(2),y_(2)) makes an angle theta with the positiv direcction of the x-axis prove that , x_(2)=x_(1)+rcostheta and y_(2)=y_(1)+rsintheta where r=overline(PQ) .

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)) .

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 .

The coordinates of any point on the line-segment joining the points ( x _(1) ,y_(1) , z_(1)) and (x_(2), y_(2),z_(2)) are ((x_(1)+kx_(2))/(k + 1),(y _(1) + ky_(2))/(k + 1),(z_(1)+kz_(2))/(k + 1)) , then the value of k will be _

Find the scalar components and magnitude of the vector joining the points P(x_(1),y_(1),z_(1))andQ(x_(2),y_(2),z_(2)) .