91Ó°ÊÓ

Two unequal parabolas have a common axis and cancavities in opposite direction. If any line parallel to the common axis meets them in \(P\) and \(Q\). Prove that the locus of the mid-point of \(P Q\) is another parabola.

If the normals at two points \(P\) and \(Q\) of a parabola \(y^{2}=4 a x\) intersect at a third point \(R\) on the curve, then the product of ordinates of \(P\) and \(Q\) is : (a) \(4 a^{2}\) (b) \(2 a^{2}\) \(\begin{array}{ll}\text { (c) }-4 a^{2} & \text { (d) } 8 a^{2}\end{array}\)

\(P\) is a point which moves in the \(x-y\) plane, such that the point \(P\) is nearer to the centre of a square than any of the sides. The four vertices of the square are \((\pm a, \pm a)\). The region in which \(P\) will move is bounded by parts of parabolas of \(\begin{array}{ll}\text { which one has the equation : } & \text { (b) } x^{2}=a^{2}+2 \alpha y\end{array}\) (a) \(y^{2}=a^{2}+2 a x\) (c) \(y^{2}+2 \alpha x=a^{2}\) (d) none of these

Show that the locus of the centroids of equilateral triangles inscribed in the parabola \(y^{2}=4 a x\) is the parabola \(9 y^{2}-4 x a+32 a^{2}=0\).

The equation of a locus is \(y^{2}+2 a x+2 b y+c=0\). Then : (a) it is an ellipse (b) it is a parabola (c) its latus rectum \(=a\) (d) its latus rectum \(=2 a\)

A double ordinate of the parabola \(y^{2}=8 p x\) is of length \(16 p\). The angie subtended by it at the vertex of the parabola is: (a) \(\pi / 4\) (b) \(\pi / 2\) (c) \(\pi\) (d) \(\pi / 3\)

A variable chord \(P Q\) of the parabola \(y^{2}=4 x\) is drawn parallel to the line \(y=x\), 1f the parameters of the points \(P\) and \(Q\) on the parabola are \(p\) and \(q\) respectively show that \(p+q=2\) Aso show that the locus of the point of intersection of the normals at \(P\) and \(Q\) is \(2 x-y=12\).

PQ is a chord of a parabola, normal at \(P: A Q\) is drawn from the vertex \(A\); and through \(P\) a line is drawn parallel to \(A Q\) meeting the axis in \(R\). Show that \(A R\) is double the focal distance of \(P\).

The equation of a tangent to the parabola \(y^{2}=8 x\) which makes an angle 45 ' with the line \(y=3 x+5\) is : (a) \(2 x+y+1=0\) (b) \(y=2 x+1\) (c) \(x-2 y+8=0\) (d) \(x+2 y-8=0\)

If the tangents drawn from the point \((0,2)\) to the parabola \(y^{2}=4 a x\) are inclined at an angle of \(\frac{3 \pi}{4}\) then the value of \(a\) is: (a) 2 (b) \(-\underline{2}\) (c) 1 (d) none of these