Condition for Perpendicularity of Two Straight Lines

Q. Let $A$ be a fixed point $(0,6)$ and $B$ be a moving point $(2t,0)$. Let $M$ be the mid-point of $AB$ and the perpendicular bisector of $AB$ meets the $y$-axis at $C$. The locus of the mid-point $P$ of $MC$ is

1. $3x^2+2y-6=0$
2. $2x^2+3y-9=0$
3. $3x^2-2y-6=0$
4. $2x^2-3y+9=0$

Q. Let $PQ$ and $RS$ be tangents at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle, then $2r$ equals

1. $\frac{2PQ\times RS}{PQ+RS}$
2. $\frac{\sqrt{(P{Q}^2+R{S}^2)}}{2}$
3. $\sqrt{PQ\times RS}$
4. $\frac{PQ+RS}{2}$

Q.

A line drawn through the point $P(-1,2)$ meets the hyperbola $xy=c^2$ at the points $A$ and $B$. (points $A$ and $B$ lie on same side of $P$) and $Q$ is a point on $AB$ such that $PA,PQ$ and $PB$ are in $H.P$ then locus of $Q$ is

A. $x-2y=2c^2$

B. $2x-y=2c^2$

C. $x+2y=2c^2$

D. $2x-y+2c^2=0$

Q. Let $PQ$ and $RS$ be tangents at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle, then $2r$ equals

1. $\sqrt{PQ\times RS}$
2. $\frac{PQ+RS}{2}$
3. $\frac{2PQ\times RS}{PQ+RS}$
4. $\frac{\sqrt{(P{Q}^2+R{S}^2)}}{2}$

Q. In a $△ABC,$ the coordinates of $A$ are $(1,10).$ If the coordinates of orthocentre and circumcentre are $(\frac{11}{3},\frac{4}{3})$ and $(-\frac{1}{3},\frac{2}{3}),$ then the side $BC$ passes through the point

1. $(-\frac{31}{3},-\frac{29}{39})$
2. $(-\frac{31}{3},\frac{29}{39})$
3. $(\frac{31}{3},\frac{31}{39})$
4. $(\frac{31}{3},-\frac{31}{39})$

Q. If the perpendicular bisector of the line segment joining the points $P(1,4)$ and $Q(k,3)$ has $y-$intercept equal to $-4,$ then a value of $k$ is

1. $-2$
2. $\sqrt{15}$
3. $\sqrt{14}$
4. $-4$

Q. The area bounded by the parabola $y^2=4x$ and the line $y=2x-4$:

1. 5 sq. units
2. 9 sq. units
3. 4 sq. units
4. 2 sq. units

Q. The normal at an end of a latus rectum of the parabola $y^2=36x$ meets the parabola again at the point

1. $(81,-54)$
2. $(81,54)$
3. $(9,18)$
4. $(9,-18)$

Q. A line $y=mx+1$ intersects the circle $(x-3{)}^2+(y+2{)}^2=25$ at the points $P$ and $Q$. If the midpoint of the line segment $PQ$ has $x$-coordinate $-\frac{3}{5}$, then which one of the following options is correct?

1. $-3\le m<-1$
2. $2\le m<4$
3. $4\le m<6$
4. $6\le m<8$

Q. A line $y=mx+1$ intersects the circle $(x-3{)}^2+(y+2{)}^2=25$ at the points $P$ and $Q$. If the midpoint of the line segment $PQ$ has $x$-coordinate $-\frac{3}{5}$, then which one of the following options is correct?

1. $-3\le m<-1$
2. $2\le m<4$
3. $4\le m<6$
4. $6\le m<8$

Q. Let $PQ$ and $RS$ be tangents at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle, then $2r$ equals

1. $\sqrt{PQ\times RS}$
2. $\frac{PQ+RS}{2}$
3. $\frac{2PQ\times RS}{PQ+RS}$
4. $\frac{\sqrt{(P{Q}^2+R{S}^2)}}{2}$

Q. Circles $C_1$ and $C_2$, of radii $r$ and $R$ respectively, touch each other as shown in figure. The line $l$, which is parallel to the line joining the centres of $C_1$ and $C_2$, is tangent to $C_1$ at $P$ and intersects $C_2$ at $A,B.$ If $R^2=2r^2$, then $\angle AOB$.

1. $22\frac{1^{\circ }}{2}$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. $67\frac{1^{\circ }}{2}$

Q. A line $y=mx+1$ intersects the circle $(x-3{)}^2+(y+2{)}^2=25$ at the points $P$ and $Q$. If the midpoint of the line segment $PQ$ has $x$-coordinate $-\frac{3}{5}$, then which one of the following options is correct?

1. $-3\le m<-1$
2. $2\le m<4$
3. $4\le m<6$
4. $6\le m<8$

Q. If the focus of the parabola $x^2-ky+3=0$ is $(0,2)$ then the values of k are?

1. $\frac{3}{2}$
2. $3$
3. $4$
4. $6$