If `O` is the origin and if the coordinates of any two points `Q_1a n dQ_2` are `(x_1,y_1)a n d(x_2,y_2),` respectively, prove that `O Q_1dotO Q_2cos/_Q_1O Q_2=x_1x_2+y_1y_2dot`

If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x_1,y_1) and (x_2,y_2), respectively, then x_1=y^2 (b) x_1=y_1 y_1=y_2 (d) x_2=y_1

The coordinates of the ends of a focal chord of the parabola y^2=4a x are (x_1, y_1) and (x_2, y_2) . Then find the value of x_1x_2+y_1y_2 .

If (x+i y)(p+i q)=(x^2+y^2)i , prove that x=q ,y=pdot

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

If the chords of contact of tangents from two poinst (x_1, y_1) and (x_2, y_2) to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 are at right angles, then find the value of (x_1x_2)/(y_1y_2)dot

If the normals to the ellipse x^2/a^2+y^2/b^2= 1 at the points (X_1, y_1), (x_2, y_2) and (x_3, y_3) are concurrent, prove that |(x_1,y_1,x_1y_1),(x_2,y_2,x_2y_2),(x_3,y_3,x_3y_3)|=0 .

The coordinates of two points Pa n dQ are (x_1,y_1)a n d(x_2,y_2)a n dO is the origin. If the circles are described on O Pa n dO Q as diameters, then the length of their common chord is (a) (|x_1y_2+x_2y_1|)/(P Q) (b) (|x_1y_2-x_2y_1|)/(P Q) (|x_1x_2+y_1y_2|)/(P Q) (d) (|x_1x_2-y_1y_2|)/(P Q)

Statement 1 :The circles x^2+y^2+2p x+r=0 and x^2+y^2+2q y+r=0 touch if 1/(p^2)+1/(q^2)=1/r dot Statement 2 : Two centers C_1a n dC_2 and radii r_1a n dr_2, respectively, touch each other if |r_1+-r_2|=c_1c_2dot

The equation to the chord joining two points (x_1,y_1) and (x_2,y_2) on the rectangular hyperbola xy=c^2 is: (A) x/(x_1+x_2)+y/(y_1+y_2)=1 (B) x/(x_1-x_2)+y/(y_1-y_2)=1 (C) x/(y_1+y_2)+y/(x_1+x_2)=1 (D) x/(y_1-y_2)+y/(x_1-x_2)=1

If the point (x_1+t(x_2-x_1),y_1+t(y_2-y_1)) divides the join of (x_1,y_1) and (x_2, y_2) internally, then t 1 (d) t=1