3d planes practice · Easy

Option A (23) is correct because by finding the intersection point P of the line and the plane, and the intersection point Q of the line and the YZ-plane, the distance PQ can be calculated and expressed in the form a$\sqrt{b}$, leading to a+b = 23.

Option A ($20 x+23 y+26 z-69=0$) is correct because it represents the equation of a plane passing through the intersection of two given planes and a specific point.

Option C ($\frac{{x-1}}{{1}}=\frac{{y-1}}{{-2}}=\frac{{z+1}}{{-3}}$) is correct because the direction ratios of the line are the same as the direction ratios of the normal to the plane.

Option C ($3 x+2 y=0$) is correct because a plane containing the z-axis must have an equation independent of z.

Option D (5) is correct because after finding the direction vector of the line and the foot of the perpendicular, the sum of the coordinates is calculated correctly.

Option D (-4) is correct because it's the sum of the x, y, and z intercepts of the plane that passes through the given point and contains the given line.

Option A ($\frac{9}{2}$) is correct because after finding the point of intersection P of the line and the plane, its distance from the origin is $\frac{9}{2}$.

Option C ($\sqrt {38}$) is correct because it is the calculated distance between the given point and the intersection point of the line and plane.

Option D ($x^2+y^2+z^2+5xy+3yz+3zx=0$) is correct because it represents the locus of a point that satisfies the given distance condition.

Option C ($3 x+4 y-5 z-9=0$) is the correct equation of the plane because it satisfies the given conditions of passing through the points and being perpendicular to the specified plane.

Option B ($2 x+3 y-z=14$) is correct because the direction ratios of OP are proportional to (2, 3, -1), and the plane passes through P(2, 3, -1).

Option C ($23x+29y+3z=55$) is correct because it uses the prime factors of 2001 as direction ratios and passes through the given point.

Option B ($(3,0,4)$) is correct because it does not satisfy the equation of the plane derived from the given foot of the perpendicular from the origin, and the condition that $a \in N$.

Option B ($10 \sqrt{3}$) is correct because it is the distance calculated along the given line from the point to the plane.

Option C ($\frac{{32}}{{19}}$) is correct because it is a value of $\alpha$ that satisfies the condition for the shortest distance between the lines.

Option D (9) is correct because it is derived from the relationship between the distance of the plane from the origin and the coordinates of the centroid of the triangle formed by the intercepts.

Option A ($x+y+z=9$) is the correct equation of the plane because it passes through the midpoint of the line segment and is perpendicular to it.

Option B ($\frac{10}{\sqrt{83}}$ units) is correct because it represents the distance of the point from the plane, calculated using the normal vector and the plane equation.

Option C ($\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} + \frac{1}{{{c^2}}} - \frac{1}{{{{a'}^2}}} - \frac{1}{{{{b'}^2}}} - \frac{1}{{{{c'}^2}}} = 0$) is correct because the perpendicular distance from the origin to the plane is the same in both coordinate systems.

Option B (0, 1, 1) is correct because it satisfies the equation of the plane $P$ which contains the given line and is perpendicular to the given plane.

Option C ($\frac{{1}}{{2}}(a^2 b^2 + b^2 c^2 + c^2 a^2)^{{1/2}}$) is correct because it represents the area of the triangle $ABC$ formed by the intercepts of the plane on the coordinate axes.

Option C ($\pm \sqrt{5}$) is correct because substituting z=0 into the line equation and solving for x and y gives x=5 and y=1, which when substituted into xy = c² yields c = $\pm \sqrt{5}$.

Option B is correct because the equation of the plane is found using the cross product of the normal vectors of the given planes and the given point.

Option A ($6 x+3 y+2 z=0$) is correct because the direction ratios of the line are proportional to 6, 3, and 2, and the plane passes through the origin.

Option C (-41 or 43) is correct because it uses the formula for the distance between two parallel planes to find the possible values of d.

Option C (5) is correct because it involves finding the equation of the plane and then calculating the distance from a point to that plane.

Option B ($\sqrt{26}$) is correct because it is the distance calculated between point A and the plane P, measured parallel to the given line.

Option A (1) is correct because the equation is already in the normal form, where the right-hand side represents the distance from the origin.

Option C ($\frac{7}{2}$) is correct because it is the distance between the two parallel planes, calculated using the formula for the distance between parallel planes.

Option C ($2 x-y+10 z=11$) is correct because it represents the equation of the plane that passes through the line of intersection of the given planes and is perpendicular to the first plane.