3d planes practice · Easy

Option B ($2 a / \sqrt{3}$) is correct because it represents the shortest distance between the skew lines DC and AB after folding the square.

Option C ($\frac{{45}}{{7}}$) is correct because it is the value of $\lambda$ obtained by using the formula for the angle between a line and a plane.

The planes $P_1, P_2,$ and $P_3$ pass through one line if the determinant of the coefficients is zero, which leads to the condition $a^2+b^2+c^2=1$.

Option B ($\frac{10}{3 \sqrt{3}}$) is the correct distance because the line is parallel to the plane, and the distance is calculated using the perpendicular distance from a point on the line to the plane.

Option B ($(5 x+6 y-12)^2+4(3 x+5 y-9)^2=1, z=6-2 x -3 y$) is correct because it represents the locus of the foot of the perpendicular from a point on the circle to the given plane.

Option D ($\sqrt{\alpha^2+\gamma^2}$) is correct because the distance from a point to the y-axis is calculated using the x and z coordinates.

The correct answer is C. The equation of the plane is found using the normal vector from the line and the given point.

The correct answer is A ($\frac{{1}}{{x^2}}+\frac{{1}}{{y^2}}+\frac{{1}}{{z^2}}=1$) because it is derived from the distance formula from the origin to the plane and the centroid coordinates.

Option C ($\sqrt{\frac{3}{2}}$) is correct because it represents the perpendicular distance from the origin to the plane, calculated using the formula for distance from a point to a plane.

Option A (4) is correct because the lengths of the edges of the parallelepiped formed are 8, 6, and 2, and 4 is not among these lengths.

Option D ($a=0$) is correct because the points (1, a, 1) and (-3, 0, a) must lie on opposite sides of the plane, and the condition is satisfied when a=0.

Option A (1) is correct because the distance is found by parameterizing the line, finding the intersection with the plane, and then calculating the distance between the points.

Option D (None of these) is correct because the calculated perpendicular distance from the origin to the plane is 4, which is not among the given options.

Option A ($y-3z+6=0$) is correct because the equation of the plane is found by taking a linear combination of the two planes and then using the condition that it is parallel to the X-axis.

Option D ($\frac{250}{83}$) is correct because it is the square of the distance calculated by finding the plane equation and using the point-to-plane distance formula.

Option C ($P_1$ and $P_3$) is correct because their normal vectors are parallel, but the planes are not identical.

Option A (3) is correct because the plane containing the x-axis and the point (1,1,1) is y-z=0. The line is parallel to the plane if the dot product of the direction vector of the line and the normal vector of the plane is zero, which leads to k = -3, and |k| = 3.

Option A ((3,-1,1)) is correct because substituting the parametric form of the line into the plane equation yields the intersection point (3,-1,1).

Option C ($\left(\frac{{a}}{{2}}, \frac{{b}}{{2}}, \frac{{c}}{{2}}\right)$) is correct because it is derived by equating the distances from the point to each of the given points.

Option D (13) is correct because the distance is calculated by finding the intersection point of the line and plane, then using the distance formula.

Option B (Assertion is correct, reason is correct; reason is not a correct explanation for assertion) is correct because both statements are true, but the reason doesn't explain the assertion.

Option D (3x – z = 0) is correct. The equation of a plane containing the y-axis and passing through (1,2,3) is found by setting up the general equation and using the given conditions.

Option B ((1,0,5)) is correct because the plane containing the given line and parallel to another line passes through this point.

Option A ($\left(\frac{5}{3},-\frac{1}{3}, 0\right)$) is correct because finding the intersection of the line passing through (0,0,1) and the given lines with the xy-plane results in this point.

Option A ($z-x=0$) is correct because the equation of the plane passing through the intersection of the X-Y and Y-Z planes and rotated by 45 degrees is $z = x$ or $z = -x$.

Option B (3/2) is correct because the condition for a plane to be parallel to a line is that the dot product of the normal vector of the plane and the direction vector of the line is zero.

Option B (greater than 3 but less than 4) is correct because the calculated perpendicular distance from the point to the line falls within this range.

Option A ($b c x+c a y-a b z=0$) is correct because it represents the equation of a plane passing through the origin and containing points A(0, b, c) and B(a, 0, c).

Option A ($Y O Z$) is correct because the equation $by + cz + d = 0$ represents a plane that does not involve x, making it perpendicular to the YOZ plane (x=0).

Option B ($\left(\frac{1}{5},-\frac{8}{15}\right)$) is correct because it satisfies the condition for the three planes to contain a common line.