Let AB be the rod making an angle \(θ\) with OX and P (x, y) be the point on it such that AP = 3 cm.
Then, \(PB = AB - AP = (12 - 3) cm = 9 cm [AB = 12 cm] \)
From P, draw \(PQ⊥OY\) and \(PR⊥OX.\)
In \( \triangle PBQ, cos θ = \frac{PQ}{PB} = \frac{x}{9}\)
In \(\triangle PRA, Sin θ =\frac{ PR}{PA} =\frac{ y}{3}\)
Since, \(sin^2 θ +cos^2 θ = 1,\)
\((\frac{y}{3})^2 + (\frac{x}{9})^2 = 1\)
or \(\frac{x^2}{81} + \frac{y^2}{9} = 1\)
Thus, the equation of the locus of point P on the rod is \(\frac{x^2}{81} + \frac{y^2}{9} = 1\)
If a tangent to the hyperbola \( x^2 - \frac{y^2}{3} = 1 \) is also a tangent to the parabola \( y^2 = 8x \), then the equation of such tangent with the positive slope is:
If a circle of radius 4 cm passes through the foci of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and is concentric with the hyperbola, then the eccentricity of the conjugate hyperbola of that hyperbola is:
Find the mean and variance for the following frequency distribution.
Classes | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
Frequencies | 5 | 8 | 15 | 16 | 6 |
When a plane intersects a cone in multiple sections, several types of curves are obtained. These curves can be a circle, an ellipse, a parabola, and a hyperbola. When a plane cuts the cone other than the vertex then the following situations may occur:
Let ‘β’ is the angle made by the plane with the vertical axis of the cone
Read More: Conic Sections