Analytical Geometry Practice Problems

In this page , you will find Analytical Geometry Practice Problems. The problems are listed below.

Q1: To what point the origin is to be moved so that one can get rid of the first degree terms from the equation

$x^2+xy+2y^2-7x-5y+12=0$.

Q2: The gradient of one of the straight lines of $ax^2+2hxy+by^2=0$ is twice that of the other. Show that $8h^2=9ab$.

Q3: Find the value of $\lambda$ so that the equation

$6x^2+xy+\lambda y^2+2x-31y-20=0$

may represent a pair of straight lines.

Q4: Prove that the pair of straight lines joining the origin to the points of intersection of the parabola $y^2=4ax$ by the straight line $y=mx+c$ are

1. at right angles, if $c+4am=0$ and
2. coincident, if $c=\dfrac{a}{m}$.

Q5: Find the centre and the radius of the sphere $3x^2+3y^2+3z^2+2x-4y-2z-1=0$.

Q6: Find the point where the straight line joining the points (2, −3,1) and (3, −4, −5) cuts the plane 3𝑥 + 𝑦 + 𝑧 = 8.

Q7: Find the equation of the plane passing through the three points (2,2, −1), (3,4,2) and (7,0,6).

Q8: Find the equation of the plane through the point (1,2,3) parallel to the plane 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 0.