Orthocentre

Q.

Find the value of k, if area of triangle is 4 sq unit and vertices are

$(k,0),(4,0)(0,2)$

$(-2,0),(0,4),(0,k)$

Q.

If area of a triangle is 35 sq unit with vertices (2, -6), (5, 4) and (k, 4), then k is
a) 12
b) -2
c) -12, -2
d) 12, -2

Q.

If $\left(1,5,35\right),\left(7,5,5\right),\left(1,\lambda ,7\right)$ and $\left(2\lambda ,1,2\right)$ are co-planar, then the sum of all possible values of $\lambda$ is:

1. $-\frac{44}{5}$

2. $\frac{39}{5}$

3. $-\frac{39}{5}$

4. $\frac{44}{5}$

Q. The orthocenter of the triangle formed by the lines x + y = 1, 2x + 3y = 6 and 4x - y + 4 = 0 lies in

1. I quadrant
2. II quadrant
3. III quadrant
4. IV quadrant

Q.

The distance between the orthocentre and circumcentre of the triangle with vertices $(1,2)(2,1)$ and $(\frac{3+\sqrt{3}}{2},\frac{3+\sqrt{3}}{2})$ is

1. $\sqrt{2}$

2. 0

3. $3+\sqrt{3}$

4. none of these

Q. Let $A(a,0),B(b,2b+1)$ and $C(0,b),b\ne 0,|b|\ne 1,$ be points such that the area of triangle $ABC$ is $1$ sq. unit, then the sum of all possible values of $a$ is

1. $\frac{2b}{b+1}$
2. $\frac{-2b^2}{b+1}$
3. $\frac{2b^2}{b+1}$
4. $\frac{-2b}{b+1}$

Q. Orthocentre of a triangle with vertices (0, 0), (3, 4) and (4, 0) is

1. $(3,\frac{5}{4})$
2. $(3,\frac{3}{4})$
3. (3, 9)
4. (3, 12)

Q. Vertices of a variable triangle are (3, 4), (5 cos $\theta$, 5 sin $\theta$) and (5 sin $\theta$, -5 cos $\theta$) where $\theta$ $\in$ R.Then locus of it's orthocenter is

1. + = 100
2. + = 100
3. + = 100
4. + = 100

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. 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. In a $△OAB$, where $O$ is origin and vertex $B$ is $(3,4)$. If the orthocentre of the triangle is $P(1,4)$ and coordinates of vertex $A$ is $(h,k)$, then the value of $4k+h$ is

Q.

The diagonal passing through origin of a quadrilateral formed by $x=0,y=0,x+y=1$ and$6x+y=3,$ is

1. $3x–2y=0$

2. $2x–3y=0$

3. $3x+2y=0$

4. None of these

Q.

A straight line through the point $(1,1)$ meets the $X-$axis at $A$ and $Y-$axis at $B$. The locus of the midpoint of $AB$ is

1. $2xy+x+y=0$

2. $x+y–2xy=0$

3. $x+y+2=0$

4. $x+y-2=0$

Q.

The equation of the line which is such that the portion of line segment intercepted between the coordinate axes is bisected at $(4,-3)$, is

1. $3x+4y=24$

2. $3x–4y=12$

3. $3x–4y=24$

4. $4x–3y=24$

Q. ABC is a triangle in which angle C is a right angle. If the coordinates of A and B be (–3, 4) and (3, –4) respectively, then the equation of the circum circle of triangle ABC is

1. 
2. 
3. 
4. None of these

Q. Let $F_1(x_1,0)$ and $F_2(x_2,0),$ for $x_1<0$ and $x_2>0,$ be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1.$ Suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.

The orthocentre of the triangle $F_{1}MN$ is

1. $(-\frac{9}{10},0)$
2. $(\frac{2}{3},0)$
3. $(\frac{9}{10},0)$
4. $(\frac{2}{3},\sqrt{6})$

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. 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. If $A(7,9),B(3,-7)$ and $C(-3,3)$ represent the vertices of a triangle, then which of the following is/are correct?

1. They form an equilateral triangle.
2. They form a right angled isosceles triangle.
3. Orthocentre of the triangle is $(-3,3)$
4. Orthocentre of the triangle is $(3,-7)$

Q.

A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle $OPQ$, where $O$ is the origin, if the area of the $OPQ$ is least, then the slope of the line $PQ$ is?

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. If $(k,2),(2,4)$ and $(3,2)$ are vertices of the triangle of area $4$ square units, then the value of $k$ is/are

1. $7$
2. $4$
3. $-1$
4. $-2$

Q.

Find the coordinates of the orthocentre of the triangles whose angular points are $(1,0),(2,-4)$, and $(-5,-2)$.

Q. If two vertices of a triangle are $(5,-1)$ and $(-2,3)$ and if its orthocentre lies at the origin, then the coordinates of the third vertex are

1. $(4,7)$
2. $(-4,-7)$
3. $(2,-3)$
4. $(5,-1)$

Q.

The value of $\left[(a-b)(b-c)(c-a)\right]$ is

1. $0$

2. $1$

3. $2\left[abc\right]$

4. $2$

Q.

Let k be an integer such that the triangle with vertices (k, -3k), (5, k) and (-k, 2) has area 28 sq units. Then, the orthocentre of this triangle is at the point

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

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. The orthocentre of the triangle formed by the vertices $(5,0),(0,0)$ and $(\frac{5}{2},\frac{5\sqrt{3}}{2})$ is

1. $(2,3)$
2. $(\frac{5}{2},\frac{5}{2\sqrt{3}})$
3. $(\frac{5}{6},\frac{5}{2\sqrt{3}})$
4. $(\frac{5}{2},\frac{5}{\sqrt{3}})$

Q.

Write the coordinates of the orthocentre of the triangle by points (8, 0), (4, 6) and (0, 0).

Q. In $△PQR,PQ=13,QR=14$ and $PR=15$. A altitude is drawn from $P$ and a parallel line is drawn as $LM$ which divides the triangle into two equal areas, then the length of $PL$ is

1. $5$
2. $5(3-\sqrt{6})$
3. $5(3-\sqrt{7})$
4. $5(3-\sqrt{8})$