Orthocentre

Q.

If a and b are real numbers between 0 and 1 such that the points $z_1$=a+i, $z_2$=1+bi and $z_3$=0 form an equilateral triangle, then

1. 

2. 

3. 

4. None of these

Q.

The orthocentre of any triangle formed by three tangents of parabola lies on the __ of the parabola.

Q.

The points (a, b), (c, d) and $(\frac{kc+la}{k+l},\frac{kd+lb}{k+l})$ are

1. Vertices of a right angled triangle

2. Vertices of an isosceles triangle

3. Collinear

4. Vertices of an equilateral triangle

Q.

O is the origin and A is the point (3, 4). If a point P moves so that the line segment OP is always

parallel to the line segment OA, then the equation to the locus of P is

1. 

2. 

3. 

4. 

Q. If the equal sides AB and AC (each equal to a) of a right angled isosceles $\Delta$ABC be produced to P and Q so that BP. CQ = ${AB}^2$, then the line PQ always passes through the fixed point.

1. (a, a)
2. (a, 0)
3. (0, a)
4. (a, 2a)

Q.

If arg $(z-1)$ =$\frac{\pi }{4}$, then the complex number z lies on a

1. circle

2. straight line

3. ellipse

4. none of these

Q.

If the vertices of a triangle be $(a{m}_1^2,2a{m}_1),(a{m}_2^2,2a{m}_2)and(a{m}_3^2,2a{m}_3)$, then the area of the triangle is

1. 

2. 

3. 

4. None of these

Q. If $\left|z\right|=1$ and $\frac{z_1-z_3}{z_2-z_3}=\frac{z-i}{z+i}$ then $z_1,z_2,z_3$ will be vertices of a

1. acute triangle
2. obtuse triangle
3. right triangle

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

Q.

(3, -1) is the image of the point P (-3, 5) about the line ax + by + c = 0 find the value of $\frac{-a}{b}$

__

Q.

Which of the following statements are true and which are false? In each case give a valid reason for your answer.

(i) p: $\sqrt{11}$ is an irrational number.

(ii) q: Circle is a particular case of an ellipse.

(iii) r: Each radius of a circle is a chord of the circle.

(iv) s: The centre of a circle bisects each chord of the circle.

(v) t: If a nd b are integers such that $a<b$, then $-a<-b.$

(vi) u: The quadratic equation $x^2+x+1=0$ has no real roots.

Q.

The orthocentre of the triangle formed by the lines xy = 0 and x + y = 1 is

1. 

2. 

3. 

4. 

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. 2sq. unit25bs(2)0(p)26. If (h, k) is orthocentre of triangle formed by lines 3y3x 2y and x+7, then (h- k) is(e)0 (9)I ()If a vertex of an equilateral triangle is the origin and theside opposite to it has the equation x + y = 1 then theorthocentre of the triangle is:(p)Sls(B)(4)(0)(d) None of these

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. Locus of the point of intersection of lines $x\cos \alpha +y\sin \alpha =a$ and $x\sin \alpha -y\cos \alpha =a$ ($\alpha ϵR$) is ?

1. $x^2+y^2=a^2$
2. $x^2+y^2=2a^2$
3. $x^2+y^2+2x+2y=a^2$
4. $Noneofthese$

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. Find the eccentricity, centre, foci, vertices of the ellipse $16x^2+9y^2+32x-36y=92$ and also draw the diagram.

Q. The equation of base of an equilateral triangle js x+y=2 and tge vertex is (2, -1). The length o side of the triangle 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. $\begin{array}{cc}Column1& Column2\\ \text{a. Two vertices of a triangle are}(5,-1)\text{and}(-2,3).& \text{p.}(-4,-7)\\ \text{If orthocenter}\text{is}\text{the origin then coordinates of}& \\ \text{the third vertex is},& \\ \text{b. A point on the line}x+y=4\text{which lies at a unit}& \text{q.}(-7,-11)\\ \text{distance from the line}4x+3y=10\text{is}& \\ \text{c. Orthocentre of the triangle formed by the lines}& \text{r.}(2,-2)\\ x+y-1=0,x-y+3=0,2x+y=7\text{is}& \\ \text{d. If}2a,b,c\text{are in A.P., then lines}ax+by+c=0\text{are}& \text{s.}(-1,2)\\ \text{concurrent at}& \end{array}$
Then which of the following is correct ?

1. $a\to s,b\to r,c\to p,d\to q$
2. $a\to p,b\to q,c\to s,d\to r$
3. $a\to q,b\to p,c\to r,d\to s$
4. $a\to r,b\to s,c\to p,d\to q$

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

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

Q.

The orthocentre of the triangle formed by the lines xy = 0 and x + y = 1 is

1. (0, 0)

2. $(\frac{1}{2},\frac{1}{2})$

3. $(\frac{1}{3},\frac{1}{3})$

4. $(\frac{1}{4},\frac{1}{4})$

Q. Find the points on the $x-axis$ such that their perpendicular distance from the line $\frac{x}{a}+\frac{y}{b}=1$ is $a,a,b>0$.

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. 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. if three distinct normlas can be drawn to the parabola y^2-2y=4x-9 from point 2a b then find range of a