The largest value of a, for which the perpendicular distance of the plane containing the lines\( \vec{r} (\hat{i}+\hat{j})+λ(\hat{i}+a\hat{j}−\hat{k})\ and\ \vec{r}=(\hat{i}+\hat{j})+μ(−\hat{i}+\hat{j}−a\hat{k}) \)from the point (2, 1, 4) is √3, is _____________.
Let the tangent drawn to the parabola y2 = 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola\(\frac{x^2}{α^2}−\frac{y^2}{β^2}=1\)at the point (α + 4, β + 4) does NOT pass through the point
Let a curve y = y(x) pass through the point (3, 3) and the area of the region under this curve, above the x-axis and between the abscissae 3 and \(x(>3)\ be\ (\frac{y}{x})^3\). If this curve also passes through the point (α,6√10) in the first quadrant, then α is equal to _______.
An ellipse\(E:\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)passes through the vertices of the hyperbola\(H:\frac{x^2}{49} - \frac{y^2}{64} = -1\)Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be 1/2. If the length of the latus rectum of the ellipse E, then the value of 113l is equal to _____.
Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = _______.
Let n ≥ 5 be an integer. If 9n – 8n – 1 = 64α and 6n – 5n – 1 = 25β, then α – β is equal to
Let the solution curve y = y(x) of the differential equation (4 + x2)dy – 2x(x2 + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _______.
If \(\frac{1}{2\times 3 \times 4} + \frac{1}{3\times 4 \times 5 } + \frac{1}{4 \times 5 \times 6 }+ \dots + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}\) then 34 k is equal to ____________.
Let P be the plane passing through the intersection of the planes
r→.(i+3k−k)=5 and r→ .(2i−j+k)=3,
and the point (2, 1, –2). Let the position vectors of the points X and Y be
i−2j+4k and 5i−j+2k
respectively. Then the points