3d planes practice · Mixed difficulty

Option D ($r \cdot (\hat i - \hat k) + 2 = 0$) is correct because it represents the equation of a plane that passes through the intersection of the given planes and is perpendicular to the plane $x - y + z = 0$.

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 ($\mathbf{r} \cdot(-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=0$) is correct because the normal vector to the plane is found by taking the cross product of the direction vectors of the lines, and the plane passes through the given point.

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.

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 A ($\frac{{x-1}}{{-3}}=\frac{{y-2}}{{-2}}=\frac{{z-3}}{{4}}$) is correct because the direction ratios of the line are found by taking the cross product of the normal vectors of the planes, and the line passes through the given point.

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 A ($\frac{x-3}{1}=\frac{y-3}{5}=\frac{z-2}{10}$) because it represents the equation of the reflected ray after finding the reflection point and the intersection point.

Option A (65) is correct because after finding the normal vector and equation of the plane, the image point Q is calculated, and then the sum of squares of its coordinates is found to be 65.

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 B (14) is correct because it is the sum of the coefficients a, b, and c of the plane equation after finding the plane parallel to the given line and passing through the intersection of the two planes.

Option A ($\frac{x}{1}=\frac{y-5}{-2}=\frac{z+2}{1}$) is correct because it represents the line of intersection between the plane perpendicular to (1,1,1) passing through P(1,1,1) and the plane x+2y+3z=4.

Option B (600) is the correct answer because it represents the volume of the tetrahedron formed by the plane and the coordinate planes, calculated using the intercepts.

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 A (32) is correct. By using the given angle between the plane and the line, we find 'a'. Then, using the distance formula, we find 'b'. Finally, we calculate a⁴ + b².

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 A (7) is correct because finding the equation of the plane passing through the intersection and parallel to the given plane yields k=7.

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

Option D (3654) is correct because by setting up ratios of consecutive binomial coefficients and solving for n and r, the coefficient of the fourth term can be calculated as 3654.

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.