Analytic Geometry Answers

Show that the plane 2𝑥 + 𝑦 + 2𝑧 = 0 is a tangent plane to the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 2𝑥 + 2𝑦 − 2𝑧 + 2 = 0.

Find the distance of the origin from the plane which passes through (2, 1, 8) , (1, 0, 2) and (−3, 4, 6)

Find the equation of the plane which passes through the line of intersection of the planes 3𝑥 + 4𝑦 − 5𝑧 = 9 and 2𝑥 + 6𝑦 + 6𝑧 = 7 and which is perpendicular to the plane 3𝑥 + 2𝑦 − 5𝑧 + 6 = 0

Find the equations of the line through (1,3, 4 ) and parallel to the line joining the points (−4, 5, 3) and (8, 9, 7)

Prove that the length of the chord of a parabola which passes through the focus and which is inclined at 30° to the axis of the parabola is four times the length of the latus rectum.

Prove that the equation of a line through (𝑥₁ , 𝑦₁) and (𝑥₂ , 𝑦₂) can be expressed in the form determinant

x y 1

x₁ y₁ 1

x₂ y₂ 1

= 0

Let 𝑃 be the midpoint of the line segment joining the points 𝐴(𝑎 + 𝑏, 𝑏) and 𝐵(𝑎 − 𝑏, 𝑎 + 𝑏). Find the slope of the line passing through 𝑃 and 𝑄 (𝑏, −𝑎/2) .

Prove that the conic passing through the points of intersection of two rectangular hyperbolas is also a rectangular hyperbola.