Slope Form a Line

Q.

Let the lines $(2–i)z=(2+i)$ and $(2+i)z+(i–2)–4i=0$, (here $i^2=–1$) be normal to a circle $C$. If the line$iz+1+i=0$ is tangent to this circle $C$, then its radius is :

1. $\frac{3}{\sqrt{2}}$

2. $3\sqrt{2}$

3. $\frac{3}{2\sqrt{2}}$

4. $\frac{1}{2\sqrt{2}}$

Q.

The equation of a circle which touches the lines $x=y$ at origin and passes through the point $\left(2,1\right)$ is $x^2+y^2+px+qy=0$ Then the value of $p,q$ are

1. $5,5$

2. $-5,5$

3. $5,-5$

4. None of these

Q.

An equilateral triangle inscribed in the circle $x^2+y^2+2gx+2fy+c$ = 0 then its side is

1. 

2. 

3. 

4. 

Q.

A straight line has its extremities on two fixed straight lines and cuts off from them a triangle of constant area of $2c^2$. Then, the locus of the middle point of the line is _____

1. 

2. 

3. 

4. 

Q. The coordinates of a point on the parabola $y^2=8x$, whose focal distance is $4$ units, is/are

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

Q. If $(h,k)$ is a point on the axis of the parabola $2(x-1{)}^2+2(y-1{)}^2=(x+y+2{)}^2$ from where three distinct normals may be drawn, then

1. $h>2$
2. $h>4$
3. $h>8$
4. $h<8$

Q. If two normals to a parabola $y^2=4ax$ intersect at right angles, then the chord joining their feet passes through a fixed point whose coordinates are

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

Q. Let a line $y=mx(m>0)$ intersect the parabola $y^2=x$ at a point $P$, other than the origin. The tangent to it at $P$ meet the $x-$ axis at point $Q$. If area $△OPQ=4\text{sq.units}$, then $2m$ is equal to

Q. Normals are drawn from the point $P$ with slopes $m_1\cdot m_2=\alpha$. If $P$ lies on the parabola $y^2=4x$ itself, then $\alpha$ is equal to

Q. If a tangent to the parabola $y^2=4ax$ makes an angle of $\frac{\pi }{3}$ with the axis of symmetry of the parabola, then point of contact(s) is/are

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