Orthocentre

Q.

In a triangle $PQR,$the coordinates of the points $P$ and $Q$ are $(-2,4)$ and $(4,2)$ respectively. If the equation of the perpendicular bisector of $PR$is $2x-y+2=0,$ then what is the circumcenter of the $∆PQR$

1. $(-2,-2)$

2. $(0,2)$

3. $(-1,0)$

4. $(1,4)$

Q.

Let$L$ be a line obtained from the intersection of two planes$x+2y+z=6$ and$y+2z=4.$ If point $P(ɑ,\beta ,\gamma )$is the foot of the perpendicular from$(3,2,1)$ on $L$, then the value of$21(ɑ+\beta +\gamma )$ equal:

1. $142$

2. $68$

3. $136$

4. $102$

Q.

A line in the $xy-$plane passes through the origin and has a slope of $\frac{1}{7}$

Which of the following points lies on the line?

1. $(0,7)$

2. $(1,7)$

3. $(7,7)$

4. $(14,2)$

5. $(7,14)$

Q.

The orthocentre of a triangle with vertices $\left(0,0\right),\left(3,4\right)$ and $\left(4,0\right)$ is

1. $\left(3,12\right)$

2. $\left(3,\frac{3}{4}\right)$

3. $\left(3,9\right)$

4. None of these

Q. Let the length of the latus rectum of an ellipse with its major-axis along $x$-axis and centre at the origin, be $8$. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

1. $(4\sqrt{3},2\sqrt{2})$
2. $(4\sqrt{2},2\sqrt{3})$
3. $(4\sqrt{3},2\sqrt{3})$
4. $(4\sqrt{2},2\sqrt{2})$

Q.

$P$ is a point on the line through a point $A$ whose position vector is$a$ and the line is parallel to the vector $b$. If $PA=6$, then the position vector of $P$ is:

1. $a+6b$

2. $a+\left(\frac{6}{\left|b\right|}\right)b$

3. $a-6b$

4. $b+\left(\frac{6}{\left|a\right|}\right)a$

Q. The distance between the orthocentre and circumcentre of a triangle whose vertices are $P(3,0),Q(0,0)$ and $R(\frac{3}{2},\frac{-3\sqrt{3}}{2})$ is

1. $0$
2. $\sqrt{2}$
3. $\sqrt{3}$
4. $3\sqrt{3}$

Q. The orthocenter of the triangle formed by (0, 0), (8, 0) and (4, 6) is

1. $4,\frac{8}{3}$
2. $3,4$
3. $4,3$
4. $-3,4$

Q. Find the angle between the tangents drawn from $x^2+y^2=41$ to an ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$

1. $0$
2. $\frac{\pi }{2}$
3. $\frac{5\pi }{6}$
4. $\frac{\pi }{3}$

Q. If $A(0,0),B(\cos \alpha ,\sin \alpha )$ and $C(\cos \beta ,\sin \beta )$ represent the vertices of a right angled triangle, then which of the following is/are correct?

1. $\sin \left(\frac{\alpha -\beta }{2}\right)=\frac{1}{\sqrt{2}}$
2. $\cos \left(\frac{\alpha -\beta }{2}\right)=-\frac{1}{\sqrt{2}}$
3. Coordinates of orthocentre are $(0,0)$
4. Coordinates of orthocentre are $(\cos \alpha ,\sin \alpha )$

Q. The orthocenter of the triangle formed by the lines $xy=0$ and $x+y=1$ is

1. $(\frac{1}{2},\frac{1}{2})$
2. $(\frac{1}{3},\frac{1}{3})$
3. $(0,0)$
4. $(\frac{1}{4},\frac{1}{4})$

Q.

Equations to the sides of a triangle are $x-3y=0$, $4x+3y=5$ and $3x+y=0$. The line $3x-4y=0$ passes through the

1. incentre

2. circumcentre

3. centroid

4. orthocentre of the triangle

Q. In $△ABC$, the length of side $AB$ is $2$ units and $\angle ABC=\frac{\pi }{3}.$ If $B$ and the mid-point of $BC$ have the coordinates $(0,0)$ and $(2,0)$ respectively, then the orthocentre of $△ABC$ is

1. $(1,3\sqrt{3})$
2. $(\frac{5}{3},\frac{1}{\sqrt{3}})$
3. $(1,\frac{1}{\sqrt{3}})$
4. $(1,\sqrt{3})$

Q. The locus of the orthocentre of the triangle formed by the lines $(1+p)x-py+p(1+p)=0$,
$(1+q)x-qy+q(1+q)=0$ and $y=0$ where $p\ne q$, is

1. A hyperbola
2. A parabola
3. An ellipse
4. A straight line

Q. If $A(7,9),B(3,-7)$ and $C(-3,3)$ represent the vertices of a triangle, then orthocentre of the triangle is

1. $(3,-7)$
2. $(-3,3)$
3. $(7,9)$
4. $(\frac{7}{3},\frac{5}{3})$

Q. The coordinates of orthocenter of a triangle formed by the coordinates O(0, 0), A(p, 0), B(0, q) are .

1. (0, 0)
2. None of these
3. $(\frac{p}{2},\frac{q}{2})$
4. $(\frac{p}{3},\frac{q}{3})$

Q. If the orthocentre of the triangle formed by the points $A(2,0),$ $B(2,3)$ and $C(0,3)$ is $(\alpha ,\beta ),$ then the value of $\alpha +\beta$ is

Q. Let the length of the latus rectum of an ellipse with its major-axis along $x$-axis and centre at the origin, be $8$. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

1. $(4\sqrt{2},2\sqrt{3})$
2. $(4\sqrt{2},2\sqrt{2})$
3. $(4\sqrt{3},2\sqrt{2})$
4. $(4\sqrt{3},2\sqrt{3})$