Conic sections - parabola, ellipse and hyperbola in standard form — practice questions

Question 1 · difficulty L2 · understanding

Suppose that two chords, drawn from the point $(1,2)$ on the circle $x^2+y^2+x-3 y=0$ are bisected by the $y$-axis. If the other ends of these chords are R and S , and the mid point of the line segment RS is $(\alpha, \beta)$, then $6(\alpha+\beta)$ is equal to :

• A. 1
• B. 3
• C. 4
• D. 6

Question 2 · difficulty L3 · understanding

If $P$ is a point on the circle $x^2+y^2=4, Q$ is a point on the straight line $5 x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of PQ , then 13 times the sum of abscissa of all such points P is $\_\_\_\_$ .

Question 3 · difficulty L3 · understanding

If one of the diameters of the circle $${x^2} + {y^2} - 2\sqrt 2 x - 6\sqrt 2 y + 14 = 0$$ is a chord of the circle $${(x - 2\sqrt 2 )^2} + {(y - 2\sqrt 2 )^2} = {r^2}$$, then the value of r 2 is equal to ____________.

Question 4 · difficulty L3 · understanding

Let the abscissae of the two points P and Q be the roots of $$2{x^2} - rx + p = 0$$ and the ordinates of P and Q be the roots of $${x^2} - sx - q = 0$$. If the equation of the circle described on PQ as diameter is $$2({x^2} + {y^2}) - 11x - 14y - 22 = 0$$, then $$2r + s - 2q + p$$ is equal to __________.

Question 5 · difficulty L3 · understanding

If one of the diameters of the circle x 2 + y 2 - 2x - 6y + 6 = 0 is a chord of another circle 'C', whose center is at (2, 1), then its radius is ________.

Question 6 · difficulty L3 · understanding

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C, whose mid-point is (1, 2), is:

• A. 4$\sqrt{2}$
• B. 2$\sqrt{2}$
• C. 2$\sqrt{3}$
• D. $\sqrt{3}$

Question 7 · difficulty L3 · understanding

Let the line x+y=1 meet the circle $x^2+y^2=4$ at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

• A. $\sqrt{14}$
• B. $3\sqrt{7}$
• C. $2\sqrt{14}$
• D. $5\sqrt{7}$

Question 8 · difficulty L3 · understanding

Let $$\mathrm{C}$$ be a circle with radius $$\sqrt{10}$$ units and centre at the origin. Let the line $$x+y=2$$ intersects the circle $$\mathrm{C}$$ at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$. Let $$\mathrm{MN}$$ be a chord of $$\mathrm{C}$$ of length 2 unit and slope $$-1$$. Then, a distance (in units) between the chord PQ and the chord $$\mathrm{MN}$$ is

• A. $$3-\sqrt{2}$$
• B. $$2-\sqrt{3}$$
• C. $$\sqrt{2}-1$$
• D. $$\sqrt{2}+1$$

Question 9 · difficulty L3 · understanding

Let the locus of the midpoints of the chords of the circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line $x+y=1$ at $\mathrm{P}$ and $\mathrm{Q}$. Then, the length of $\mathrm{PQ}$ is :

• A. $\frac{1}{2}$
• B. 1
• C. $\frac{1}{\sqrt{2}}$
• D. $\sqrt{2}$

Question 10 · difficulty L3 · understanding

A line segment AB of length $$\lambda$$ moves such that the points A and B remain on the periphery of a circle of radius $$\lambda$$. Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius :

• A. $${2 \over 3}\lambda$$
• B. $${3 \over 5}\lambda$$
• C. $${{\sqrt {19} } \over 7}\lambda$$
• D. $${{\sqrt {19} } \over 5}\lambda$$