Circle in diameter form; line-circle intersection; tangency — practice questions

Question 1 · difficulty L2 · understanding

The minimum distance between any two points P 1 and P 2 while considering point P 1 on one circle and point P 2 on the other circle for the given circles' equations x 2 + y 2 $$-$$ 10x $$-$$ 10y + 41 = 0 x 2 + y 2 $$-$$ 24x $$-$$ 10y + 160 = 0 is ___________.

Question 2 · difficulty L2 · understanding

Choose the correct statement about two circles whose equations are given below : x 2 + y 2 $$-$$ 10x $$-$$ 10y + 41 = 0 x 2 + y 2 $$-$$ 22x $$-$$ 10y + 137 = 0

• A. circles have same centre
• B. circles have no meeting point
• C. circles have only one meeting point
• D. circles have two meeting points

Question 3 · difficulty L2 · understanding

For the four circles M, N, O and P, following four equations are given : Circle M : x 2 + y 2 = 1 Circle N : x 2 + y 2 $$-$$ 2x = 0 Circle O : x 2 + y 2 $$-$$ 2x $$-$$ 2y + 1 = 0 Circle P : x 2 + y 2 $$-$$ 2y = 0 If the centre of circle M is joined with centre of the circle N, further center of circle N is joined with centre of the circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a :

• A. Rhombus
• B. Square
• C. Rectangle
• D. Parallelogram

Question 4 · difficulty L2 · understanding

Choose the incorrect statement about the two circles whose equations are given below : x 2 + y 2 $$-$$ 10x $$-$$ 10y + 41 = 0 and x 2 + y 2 $$-$$ 16x $$-$$ 10y + 80 = 0

• A. Distance between two centres is the average of radii of both the circles.
• B. Both circles pass through the centre of each other.
• C. Circles have two intersection points.
• D. Both circle's centers lie inside region of one another.

Question 5 · difficulty L3 · understanding

Let the centre of a circle, passing through the points $$(0,0),(1,0)$$ and touching the circle $$x^2+y^2=9$$, be $$(h, k)$$. Then for all possible values of the coordinates of the centre $$(h, k), 4\left(h^2+k^2\right)$$ is equal to __________.

Question 6 · difficulty L3 · understanding

Consider two circles $$C_1: x^2+y^2=25$$ and $$C_2:(x-\alpha)^2+y^2=16$$, where $$\alpha \in(5,9)$$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $$C_1$$ and $$C_2$$ be $$\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)$$. If the length of common chord of $$C_1$$ and $$C_2$$ is $$\beta$$, then the value of $$(\alpha \beta)^2$$ equals _______.

Question 7 · difficulty L3 · understanding

Equations of two diameters of a circle are $$2 x-3 y=5$$ and $$3 x-4 y=7$$. The line joining the points $$\left(-\frac{22}{7},-4\right)$$ and $$\left(-\frac{1}{7}, 3\right)$$ intersects the circle at only one point $$P(\alpha, \beta)$$. Then, $$17 \beta-\alpha$$ is equal to _________.

Question 8 · difficulty L3 · understanding

Let the mirror image of a circle $$c_{1}: x^{2}+y^{2}-2 x-6 y+\alpha=0$$ in line $$y=x+1$$ be $$c_{2}: 5 x^{2}+5 y^{2}+10 g x+10 f y+38=0$$. If $$\mathrm{r}$$ is the radius of circle $$\mathrm{c}_{2}$$, then $$\alpha+6 \mathrm{r}^{2}$$ is equal to ________.

Question 9 · difficulty L3 · understanding

If the variable line 3x + 4y = $$\alpha$$ lies between the two circles (x $$-$$ 1) 2 + (y $$-$$ 1) 2 = 1 and (x $$-$$ 9) 2 + (y $$-$$ 1) 2 = 4, without intercepting a chord on either circle, then the sum of all the integral values of $$\alpha$$ is ___________.

Question 10 · difficulty L3 · understanding

Let the set of all values of $r$, for which the circles $(x+1)^2+(y+4)^2=r^2$ and $x^2+y^2-4 x-2 y-4=0$ intersect at two distinct points be the interval $(\alpha, \beta)$. Then $\alpha \beta$ is equal to

• A. 21
• B. 24
• C. 20
• D. 25