Equation of Family of Circles Touching a Line and Passing through a Given Point on the Line

Q. A circle touches the hypotenuse of a right angled triangle at its middle point and passes through the mid point of the shorter side. If $a$ and $b$ $(a<b)$ be the length of the sides, then the radius of circle is

1. $\frac{b}{a}\sqrt{a^2+b^2}$ units
2. None of these
3. $\frac{b}{4a}\sqrt{a^2+b^2}$ units
4. $\frac{b}{2a}\sqrt{a^2+b^2}$ units

Q. The radius of the circle which touches the line $x+y=0$ at $M(-1,1)$ and cuts the circle $x^2+y^2+6x-4y+18=0$ orthogonally, is

1. $3\sqrt{2}$
2. $4\sqrt{2}$
3. $\sqrt{2}$
4. $5\sqrt{2}$

Q. If the lines $x-2y+3=0$ and $3x+ky+7=0$ cut the coordinate axes in concyclic points, then

1. $k=\frac{3}{2}$
2. $k=-\frac{3}{2}$
3. Equation of circle passing through those points is ${(x+\frac{8}{3})}^2+{(y-\frac{37}{12})}^2=\frac{1385}{12}$
4. Center of circle is $(-\frac{8}{3},\frac{37}{12})$

Q. The circle passing through $(1,-2)$ and touching the $x$-axis at $(3,0)$ also passes through the point

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

Q.

Find the equation of the circle which touches the axes and whose' centre lies on x - 2y = 3.

Q.

If $\theta$ is the angle between the tangents from $(-1,0)$ to the circle $x^2+y^2-5x+4y-2=0$, then $\theta$ is equal to:

1. $2{\tan }^{-1}\frac{7}{4}$

2. ${\tan }^{-1}\frac{7}{4}$

3. $2{\cot }^{-1}\frac{7}{4}$

4. ${\cot }^{-1}\frac{7}{4}$

Q. Let $(a,b)$ be a point on a circle which passes through $(-3,1)$ and touches the line $x+y=2$ at the point $(1,1).$ If maximum possible value of $a$ is $\alpha ,$ then a quadratic equation with rational coefficients whose one root is $\alpha ,$ is

1. $x^2+2x-9=0$
2. $x^2+2x-7=0$
3. $x^2+2x-6=0$
4. $x^2+2x-8=0$

Q.

The triangle ABC has medians AD, BE, CF . AD lies along the line y = x + 3, BE lies along the line y = 2x + 4, AB has length 60 and angle C = 90°, then the area of triangle ABC is
(A) 400 (B) 200
(C) 100 (D) none of these

Q. A circle touches the hypotenuse of a right angled triangle at its middle point and passes through the mid point of the shorter side. If $a$ and $b$ $(a<b)$ be the length of the sides, then the radius of circle is

1. $\frac{b}{a}\sqrt{a^2+b^2}$ units
2. $\frac{b}{2a}\sqrt{a^2+b^2}$ units
3. $\frac{b}{4a}\sqrt{a^2+b^2}$ units
4. None of these

Q. At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (−4, −3). Find the equation of the curve given that it passes through (−2, 1).

Q. Find the coefficient of
$x^5$ in the expansion of $(1-3x+3x^2{)}^{-\frac{1}{2}}$.

Q. Consider a family of circles which are passing through the point (–1, 1) and has x-axis as its tangents. If (h, k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval

1. 0 $<$ k $<$ $\frac{1}{2}$
2. k $\ge$ $\frac{1}{2}$
3. $-\frac{1}{2}$$\le$ k $\le$ $\frac{1}{2}$
4. k $\le$ $\frac{1}{2}$

Q. Match the following with the integrating factor of the D.E., A, B, C, D

List I	List II
A) $\frac{dy}{dx}-y\cot x=\csc x$	1. $x\sin x$
B) $x\log x\frac{dy}{dx}+y=2\log x$	2. $x{e}^x$
C) $x\sin x\frac{dy}{dx}+y(x\cos x+\sin x)=\sin x$	3. $\cos ecx$
D) $x\frac{dy}{dx}+y(1+x)=1$	4. $-\cos ecx$
	5. $logx$

1. 3, 5, 1, 4
2. 3, 5, 1, 2
3. 3, 1, 5, 4
4. 2, 3, 4, 5

Q. If the lines $x-2y+3=0$ and $3x+ky+7=0$ cut the coordinate axes in concyclic points, then

1. $k=\frac{3}{2}$
2. $k=-\frac{3}{2}$
3. Equation of circle passing through those points is ${(x+\frac{8}{3})}^2+{(y-\frac{37}{12})}^2=\frac{1385}{12}$
4. Center of circle is $(-\frac{8}{3},\frac{37}{12})$

Q.

Find the equation of circle which touches $2x-y+3=0$ and pass through the points of intersection of the line $x+2y-1=0$ and the circle $x^2+y^2-2x+1=0$

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

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

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

4. $x^2+y^2+4x+4y+3=0$

Q. Draw a rough sketch of the region {(x, y) : y² ≤ 5x, 5x² + 5y² ≤ 36} and find the area enclosed by the region using method of integration.

Q. For an isosceles triangle $ABC$ inscribed in a circle of given radius $r$ units, if $h$ is the length of altitude from vertex $A$ to $BC,$ then the value of $\underset{h\to 0}{lim}\frac{\Delta }{P^3}$ is ( where $\Delta ,P$ represent area and perimeter of triangle $ABC$ respectively)

1. $\frac{1}{r}$
2. $\frac{1}{64r}$
3. $\frac{1}{128r}$
4. $\frac{1}{2r}$

Q. The radius of the circle which touches the line $x+y=0$ at $M(-1,1)$ and cuts the circle $x^2+y^2+6x-4y+18=0$ orthogonally, is

1. $3\sqrt{2}$
2. $4\sqrt{2}$
3. $\sqrt{2}$
4. $5\sqrt{2}$

Q.

Find the equation of the circle passing the point (1, 2) and touching the line 2x + y - 1 = 0 at the point (1, -1).

1. $x^2+y^2+8x+y+y=0$

2. $x^2+y^2-4x-2y+5=0$

3. $x^2+y^2-8x-y+5=0$

4. $x^2+y^{2}4x+2y-5=0$

Q.

Find the equation of circle which touches $2x-y+3=0$ and pass through the points of intersection of the line $x+2y-1=0$ and the circle $x^2+y^2-2x+1=0$

1. 

2. 

3. 

4. 

Q.

find the equation of circle which touches 2x -y +3 = 0 and passes through the point of intersection of the line x +2y -1 =0 and the circle ^x2 +^y2 -2x + 1 = 0

Q. If $(a,b)$ are the coordinates of the center of the circle inscribed in a triangle whose vertices are $(-36,7),(20,7)$ and $(0,-8)$ then $a+b$ is equal to ____.

Q. Consider a family of circles which are passing through the point (–1, 1) and has x-axis as its tangents. If (h, k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval

1. 0 k
2. k
3. k
4. k

Q. A variable circle passes through the point $A(2018,2009)$ and touches the x axis, then the locus of the other end of the diameter through $A$ is a/an

Q.

A circle passes through the point $(3,\sqrt{\frac{7}{2}})$ and touches the line pair $x^2-y^2-2x+1=0$. The coordinates of the centre of the circle are

1. (4, 0)

2. (5, 0)

3. (6, 0)

4. none of these

Q.

A circle passes through the point $(3,\sqrt{\frac{7}{2}})$ and touches the line pair $x^2-y^2-2x+1=0$. The coordinates of the centre of the circle are

1. (5, 0)

2. (6, 0)

3. (0, 4)

4. none of these

Q. If the line y = 2 X under root 3 + k touches the circle x square + 5 square =16 then find the value of k

Q. ${\int }_{\frac{\pi }{8}}^{\frac{3\pi }{8}}\frac{cosx}{cosx+sinx}dx=$

1. $\frac{\pi }{6}$
2. $\frac{\pi }{8}$
3. $\frac{\pi }{24}$
4. $\frac{\pi }{12}$

Q. The acute angle between the common tangents of two circles $x^2+y^2=25$ and $(x-10{)}^2+y^2=100$ is

1. $60°$
2. $30°$
3. $75°$
4. $15°$

Q. In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x − x² and y = x² − x?